VARGA S THEOREM IN NUMBER FIELDS

Size: px
Start display at page:

Download "VARGA S THEOREM IN NUMBER FIELDS"

Transcription

1 VARGA S THEOREM IN NUMBER FIELDS PETE L. CLARK AND LORI D. WATSON Abstract. We give a number field version of a recent result of Varga on solutions of polynomial equations with binary input variables and relaxed output variables. 1. Introduction This note gives a contribution to the study of solution sets of systems of polynomial equations over finite local principal rings in the restricted input / relaxed output setting. The following recent result should help to explain the setting and scope. Let n, a 1,..., a n Z + and 1 N n i=1 a i. Put { 1 if N < n ma 1,..., a n ; N = min n i=1 y i if n N n i=1 a ; i the minimum is over y 1,..., y n Z n with 1 y i a i for all i and n i=1 y i = N. Theorem 1.1. [Cl18, Thm. 1.7] Let R be a Dedekind domain, and let p be a maximal ideal in R with finite residue field R/p = F q. Let n, r, v 1,..., v r Z +. Let A 1,..., A n, B 1,..., B r R be nonempty subsets each having the property that no two distinct elements are congruent modulo p. Let r, v 1,..., v r Z +. Let P 1,..., P r R[t 1,..., t n ] be nonzero polynomials, and put n za B := #{x A i 1 j m P j x B j mod p vj }. i=1 Then za B = 0 or n za B m #A 1,..., #A n ; #A i i=1 r q vj #B j degp j. Remark 1.2. For every finite local principal ring r, there is a number field K, a prime ideal p of the ring of integers Z K of K, and v Z + such that r = Z K /p v [Ne71], [BC15]. Henceforth we will work in the setting of residue rings of Z K. If in Theorem 1.1 we take v 1 = = v r = 1, A i = F q for all i and B j = {0} for all j, then we recover a result of E. Warning. Theorem 1.3. Warning s Second Theorem [Wa35] Let P 1,..., P r F q [t 1,..., t n ] be nonzero polynomials, and let z = #{x = x 1,..., x n F n q P 1 x = = P r x = 0}. Then z = 0 or z q n n degpj. 1

2 2 PETE L. CLARK AND LORI D. WATSON By Remark 1.2, we may write F q as Z K /p for a suitable maximal ideal p in the ring of integers Z K of a suitable number field K. Having done so, Theorem 1.3 can be interpreted in terms of solutions to a congruence modulo p, whereas Theorem 1.1 concerns congruences modulo powers of p. At the same time, we are restricting the input variables x 1,..., x n to lie in certain subsets A 1,..., A n and also relaxing the output variables: we do not require that P j x = 0 but only that P j x lies in a certain subset B j modulo p vj. There is however a tradeoff: Theorem 1.1 contains the hypothesis that no two elements of any A i resp. B j are congruent modulo p. Thus, whereas when v j = 1 for all j we are restricting variables by choice e.g. we could take each A i to be a complete set of coset representatives for p in Z K as done above when v j > 1 we are restricting variables by necessity we cannot take A i to be a complete set of coset representatives for p vj in Z K. We would like to have a version of Theorem 1.1 in which the A i s can be any nonempty finite subsets of Z K and the B j can be any nonempty finite subsets of Z K containing {0}. However, to do so the degree conditions need to be modified in order to take care of the arithmetic of the rings Z K /p dj. In general this seems like a difficult and worthy problem. An interesting special case was resolved in recent work of L. Varga [Va14]. His degree bound comes in terms of a new invariant of a subset B Z/p d Z \ {0} called the price of B and denoted prb that makes interesting connections to the theory of integer-valued polynomials. Theorem 1.4. Varga [Va14, Thm. 6] Let P 1,..., P r Z[t 1,..., t n ] \ {0} be polynomials without constant terms. For 1 j r, let d j Z +, and let B j Z/p dj Z be a subset containing 0. If r degp j prz/p dj Z \ B j < n, then #{x {0, 1} n 1 j r, P j x B j mod p dj } 2. In this note we will revisit and extend Varga s work. Here is our main result. Theorem 1.5. Let K be a number field of degree N, and let e 1,..., e N be a Z-basis for Z K. Let p be a nonzero prime ideal of Z K, and let d 1,..., d r Z +. Let P 1,..., P r Z K [t 1,..., t n ] be nonzero polynomials without constant terms. For each 1 j r, there are unique {ϕ j,k } 1 k N Z[t 1,..., t n ] such that N 1 P j t = ϕ j,k e j. For 1 j r, let B j be a subset of Z K /p dj that contains 0 mod p dj. Let r N S := degϕ j,k prz K /p dj \ B j. Then #{x {0, 1} n 1 j r, P j x mod p dj B j } 2 n S. Thus we extend Varga s Theorem 1.4 from Z to Z K and refine the bound on the number of solutions. In 2 we discuss the price of a subset of Z K /p d. It seems to us that Varga s definition of the price has minor technical flaws: as we understand it, he tacitly assumes that for an integer-valued polynomial f Q[t] and m, n Z, the output fn modulo m depends only on the input modulo m. This is not true: for instance if ft = tt 1 2, then fn modulo 2 depends on n modulo 4, not

3 VARGA S THEOREM IN NUMBER FIELDS 3 just modulo 2. So we take up the discussion from scratch, in the context of residue rings of Z K. The proof of Theorem 1.5 occupies 3. After setting notation in 3.1 and developing some preliminaries on multivariate Gregory-Newton expansions in 3.2, the proof proper occurs in The Price Consider the ring of integer-valued polynomials We have inclusions of rings Let IntZ K, Z K = {f K[t] fz K Z K }. Z K [t] IntZ K, Z K K[t]. mp, 0 := {f IntZ K, Z K f0 0 mod p}. Observe that mp, 0 is the kernel of a ring homomorphism IntZ K, Z K Z K /p: first evaluate f at 0 and then reduce modulo p. So mp, 0 is a maximal ideal of IntZ K, Z K. We put Up, 0 := IntZ K, Z K \ mp, 0 = {f IntZ K, Z K f0 / p}. Let d Z +, and let B be a subset of Z K /p d. We say that h Up, 0 covers B if: for all b Z K such that b mod p d B, we have hb p. The price of B, denoted prb, is the least degree of a polynomial h Up, 0 that covers B, or if there is no such polynomial. Remark 2.1. a If B 1, B 2 are subsets of Z K /p d \ {0}, then prb 1 B 2 prb 1 + prb 2 : If for i = 1, 2 the polynomial h i Up, 0 covers B i and has degree d i, then h 1 h 2 Up, 0 covers B 1 B 2 and has degree d 1 + d 2. b If 0 mod p d B, then prb = : Since 0 B we need h0 p, contradicting h Up, 0. c If d = 1, then for any subset B Z K /p \ {0}, we have prb #B: Let B be any lift of B to Z K. Then h = x Bt x Z K [t] IntZ K, Z K covers B and has degree #B. Note that here we use polynoimals with Z K -coefficients. It is clear that #B is the minimal degree of a covering polynomial h with Z K -coefficients: we can then reduce modulo p to get a polynomial in F q [t] that we want to be 0 at the points of B and nonzero at 0, so of course it must have degree at least #B. d If we assume no element of B is 0 modulo p, let B be the image of B under the natural map Z K /p d Z K /p = F q ; then our assumption gives 0 / B. Above we constructed a polynomial h Z K [t] of degree #B such that h0 / p and for all x Z K such that x mod p B, we have hx p. This same polynomial h covers B and shows that prb prb #B. For B Z K /p d \ {0} we define κb Z +, as follows. For 1 i d we will recursively define B i Z K /p i \ {0} and k i 1 N. Put B d = B, and let k d 1 be the number of elements of B d that lie in p d 1. Having defined B i and k i 1, we let B i 1 be the set of x Z K /p i 1 such that there are more than k i 1 elements of B i mapping to x under reduction modulo p i 1. We let k i 2 be the number of elements of B i 1 that lie in p i 2.

4 4 PETE L. CLARK AND LORI D. WATSON Notice that 0 / B i for all i: indeed, B i is defined as the set of elements x such that the fiber under the map Z K /p i+1 Z K /p i has more elements of B i+1 than does the fiber over 0. We put Lemma 2.2. We have κb q d 1. d 1 κb := k i q i. Proof. Each k i is a set of elements in a fiber of a q-to-1 map, so certainly k i q. In order to have k i = q, then B i+1 would need to contain the entire fiber over 0 Z K /p i, but this fiber includes 0 Z K /p i+1, which as above does not lie in B i+1. So i=0 d 1 d 1 κb = k i q i q 1q i = q d 1. i=0 Theorem 2.3. For any subset B Z K /p d \ {0}, we have prb κb. i=0 Proof. Step 1: For r 1, let A = {a 1,..., a q d 1} Z K /p d be a complete residue system modulo p d 1 none of whose elements lie in p d. We will show how to cover A with f Up, 0 of degree q d 1. We denote by v p the p-adic valuation on K. Let λ Z K be an element with v p λ = d 2 j=0 qj, and let β Z K be an element such that v p β = 0 and for all nonzero prime ideals q p of Z K, we have v q β v q λ. Such elements exist by the Chinese Remainder Theorem. Put g A t := t a j Z K [t], h A t := β λ g At K[t]. q d 1 For all x Z K, {x a 1,..., x a q d 1} is a complete residue system modulo p d 1, so in q d 1 x a j, for all 0 j d 1 there are q d 1 j factors in p j, so v p g A x d 2 j=0 qj and thus v p h A x 0. For any prime ideal q p of Z K, both v q g A x and v q β λ are non-negative, so v q h A x 0. Thus h A Int Z K. Moreover the condition that no a j lies in p d ensures that v p g A 0 = d 2 j=0 qj, so h A Up, 0. If x Z K is such that x a j mod p d for some j, then v p x a j d. Since in the above lower bounds of v p g A x we obtained a lower bound of at most d 1 on the p-adic valuation of each factor, this gives an extra divisibility and shows that v p h A x 0. Thus h A covers A with price at most q d 1. Step 2: Now let B Z K /p d \ {0}. The number of elements of B that lie in p d 1 is k d 1. For each of these elements x i we choose a complete residue system A i modulo p d 1 containing it; since no x i lies in p d this system satisfies the hypothesis of Step 1, so we can cover each A i with price at most q d 1 and thus using Remark 2.1a all of the A i s with price at most k d 1 q d 1. However, by suitably choosing the A i s we can cover many other elements as well. Indeed, because we are choosing k d 1 complete residue systems modulo p d 1, we can cover every element x that is congruent modulo p d 1 to at most k d 1 elements of B. By definition of B d 1, this means that we can cover all elements of B that do not map modulo p d 1 into B d 1. Now suppose that we can cover B d 1 by h Up, 0 of degree κ. This means that for every x Z K such that x mod p d 1 lies in B d 1, hx p. But then every element of B whose image in p d 1 lies in B d 1 is covered by h, so altogether we get prb k d 1 q d 1 + prb d 1. Now applying the same argument successively to B d 1,..., B 1 gives prb i k i 1 q i 1 + prb i 1

5 VARGA S THEOREM IN NUMBER FIELDS 5 and thus d 1 prb k i q i = κb. i=0 3. Proof of the Main Theorem 3.1. Notation. Let K be a number field of degree N, and let e 1,..., e N be a Z-basis for Z K. A Z-basis for Z K [t 1,..., t n ] is given by e j t I as j ranges over elements of {1,..., N} and I ranges over elements of N n. So for any f Z K [t 1,..., t n ], we may write 2 f = ϕ 1 t 1,..., t n e ϕ N t 1,..., t n e N, ϕ i Z[t 1,..., t n ]. Then we have For a subset B Z K /p d, we put 3.2. Multivariable Newton Expansions. deg f = max deg ϕ i. i B = Z K /p d \ B. Lemma 3.1. If f Q[t] is a polynomial and fn Z, then fz Z. Proof. See e.g. [CC, p. 2]. Theorem 3.2. Let f K[t]. a There is a unique function α f : N N K, r α r f such that i we have α r f = 0 for all but finitely many r N N, and ii for all x = x 1 e x N e N Z K, we have 3 fx = r N N α r f x1 r 1 b The following are equivalent: i We have f IntZ K, Z K. ii For all r N N, α r f Z K. We call the α r f the Gregory-Newton coefficients of f. Proof. Step 1: Let f K[t]. Let e 1,..., e N be a Z-basis for Z K. We introduce new independent indeterminates t 1,..., t N and make the substitution N t = e k t k to get a polynomial f K[t]. This polynomial induces a map K N K hence, by restriction, a map Z N K. For x = x 1,..., x N Z N, write x = x 1 e x N e N Z K. Then we have fx = fx. Let M = MapsZ N, K be the set of all such functions, and let P be the K-subspace of M consisting of functions obtained by evaluating a polynomial in K[t] on Z N, as above. By the CATS Lemma [Cl14, Thm. 12], the map K[t] P is an isomorphism of K-vector spaces. Henceforth we will identify K[t] with P inside M. xn r N.

6 6 PETE L. CLARK AND LORI D. WATSON Step 2: For all 1 k N, we define a K-linear endomorphism k of M, the kth partial difference operator: k g : x Z N gx + e k gx. These endomorphisms all commute with each other: i j g = gx + e i + e j gx + e j gx + e i + gx = j i g. Let 0 k be the identity operator on M, and for i Z+, let i k be the i-fold composition of k. For I = i 1,..., i N N N, put I = i1... i N End K M. When we apply k to a monomial t I, we get another polynomial. More precisely, if deg tk t I = 0 then k t I is the zero polynomial; otherwise deg tk k t I = deg tk t I 1; l k, deg tl k t I = deg tl t I. Thus for each f P, for all but finitely many I N N, we have that I f = 0. For the one variable difference operator, we have x x + 1 x x = =. r r r r 1 From this it follows that for I, r N N we have 4 I x1 xn = = δ r,i r 1 r 1 i 1 r N i N r N So if β : N N K is any finitely nonzero function then for all I N N we have 5 I r N N β r x1 r 1 xn r N 0 = β I, and thus there is at most one such function satisfying 3, namely α f : r r f0. So for any f M and r N N, we define the Gregory-Newton coefficient α r f := r f0 K. We may view the assignment of the package {α r f} r N N of Gregory-Newton coefficients to f M as a K-linear mapping M K NN. If we put M + = MapsN N, K, then we get a factorization M M + α K N N, where the first map restricts from Z N to N N, and the factorization occurs because the Gregory- Newton coefficients depend only on the values of f on N N. We make several observations: First Observation: The map α is an isomorphism. Indeed, knowing all the successive differences at 0 is equivalent to knowing all the values on N N, and all possible packages of Gregory-Newton coefficients arise. Namely, let S n be the assertion that for all x N N with k x k = n and all f M, then fx is a Z-linear combination of its Gregory-Newton coefficients. The case n = 0 is clear: f0 = α 0 f. Suppose S n holds for n, let x N N be such that k x k = n + 1, and choose k such that x = y + e k ; thus k y k = n. Then fx = fy + k fy.

7 VARGA S THEOREM IN NUMBER FIELDS 7 By induction, fy is a Z-linear combination of the Gregory-Newton coefficients of f and k fy is a Z-linear combination of the Gregory-Newton coefficients of k f. But every Gregory-Newton coefficient of k f is also a Gregory-Newton coefficient of f, completing the induction. Second Observation: The composite map K[t] M M + α K N N is an injection. Indeed, the kernel of M K NN is the set of functions that vanish on Z N \ N N. In particular, any element of the kernel vanishes on the infinite Cartesian subset Z <0 N and thus by the CATS Lemma is the zero polynomial. Third Observation: For a subring R K and f M, we have fn N R iff all of the Gregory-Newton coefficients of f lie in R. This is a consequence of the First Observation: the Gregory-Newton coefficients are Z-linear combinations of the values of f on N N and conversely. Step 3: For F K[t], we define the Newton expansion T F = t1 tn α r F K[t]. r 1 r N r N N This is a finite sum. Moreover, by definition of α r F and by 5 we get that for all r N N, α r T F = α r F. It now follows from Step 2 that T F = F K[t]. Applying this to the f associated to f K[t] in Step 1 completes the proof of part a. Step 4: If we assume that f IntZ K, Z K then fz N Z K so all the Gregory-Newton coefficents lie in Z K. Conversely, if all the Gregory-Newton coefficients of f lie in Z K, then for x = x 1 e x N e N Z K, by Lemma 3.1 and 3 we have fx = fx 1,..., x N Z K, so f IntZ K, Z K Proof of Theorem 1.5. We begin by recalling the following result. Theorem 3.3. Let F be a field, and let P F [t 1,..., t n ] be a polynomial. Let Then either #U = 0 or #U 2 n degp. U := {x {0, 1} n P x 0}. Proof. This is a special case of a result of Alon-Füredi [AF93, Thm. 5]. We now turn to the proof of Theorem 1.5. Put Z := {x {0, 1} n 1 j r, P j x mod p dj B j }. Step 0: If q = #Z K /p is a power of p, then we have p d p d. Therefore in 1 if we modify any coefficient of ϕ j,k t by a multiple of p d, it does not change P j modulo p d and thus does not change the set Z. We may thus assume that every coefficient of every ϕ j,k is non-negative. Step 1: For w = k i=1 tii a sum of monomials and 0 r k, we put Ψ r w := t Ii 1 t I ir. 1 i 1<i 2<...<i r k For x {0, 1} n, we have wx = #{1 i k x Ii = 1}, so wx Ψ r wx =. r For f Z K [t 1,..., t n ], write f = N ϕ kte k and suppose that all the coefficients of each ϕ k are non-negative equivalently, each ϕ k t is a sum of monomials. For r N N, we put Ψ r f := Ψ r1 ϕ 1 Ψ rn ϕ N Z[t].

8 8 PETE L. CLARK AND LORI D. WATSON For h IntZ K, Z K with Gregory-Newton coefficients α r, we put Ψ h f := r N N α r Ψ r f Z K [t]. For x {0, 1} n, we have so using 2 we get Ψ h fx = r Ψ r x = N ϕk x, α r Ψ r fx = r r k ϕ1 x ϕn x α r r 1 r N = hϕ 1 xe ϕ N xe N = hfx. Step 2: For 1 j r, let h j IntZ K, Z K have degree prb j and cover B j. Put r F := Ψ hj P j mod p Z K /p[t] = F q [t]. Note that degf r deg Ψ hj P j r n degh j degϕ j,k = S. Here is the key observation: for x {0, 1} n, if F x 0, then for all 1 j r we have p Ψ hj P j x = h j P j x, so P j x mod p dj / B j, and thus x Z. Step 3: For all 1 j r we have P j 0 = 0 and h j Up, 0, so h j 0 / p, so r r r F 0 = Ψ h j P j 0 = h j P j 0 mod p = h j 0 mod p 0. Applying Alon-Füredi to F, we get completing the proof of Theorem 1.5. #Z #{x {0, 1} n F x 0} 2 n deg F 2 n S, References [AF93] N. Alon and Z. Füredi, Covering the cube by affine hyperplanes. Eur. J. Comb , [BC15] A. Brunyate and P.L. Clark, Extending the Zolotarev-Frobenius approach to quadratic reciprocity. Ramanujan J , [CC] P.-J. Cahen and J.-L. Chabert, Integer-valued polynomials. Mathematical Surveys and Monographs, 48. American Mathematical Society, Providence, RI, [CFS14] P.L. Clark, A. Forrow and J.R. Schmitt, Warning s Second Theorem with restricted variables. Combinatorica , [Cl14] P.L. Clark, The Combinatorial Nullstellensätze Revisited. Electronic Journal of Combinatorics. Volume 21, Issue Paper #P4.15. [Cl18] P.L. Clark, Fattening up Warning s Second Theorem. pdf [Ne71] A.A. Nečaev, The structure of finite commutative rings with unity. Mat. Zametki , [Va14] L. Varga, Combinatorial Nullstellensatz modulo prime powers and the parity argument. Electron. J. Combin , no. 4, Paper 4.44, 17 pp. [Wa35] E. Warning, Bemerkung zur vorstehenden Arbeit von Herrn Chevalley. Abh. Math. Sem. Hamburg ,

VARGA S THEOREM IN NUMBER FIELDS. Pete L. Clark Department of Mathematics, University of Georgia, Athens, Georgia. Lori D.

VARGA S THEOREM IN NUMBER FIELDS. Pete L. Clark Department of Mathematics, University of Georgia, Athens, Georgia. Lori D. #A74 INTEGERS 18 (2018) VARGA S THEOREM IN NUMBER FIELDS Pete L. Clark Department of Mathematics, University of Georgia, Athens, Georgia Lori D. Watson 1 Department of Mathematics, University of Georgia,

More information

WARNING S SECOND THEOREM WITH RESTRICTED VARIABLES

WARNING S SECOND THEOREM WITH RESTRICTED VARIABLES WARNING S SECOND THEOREM WITH RESTRICTED VARIABLES PETE L. CLARK, ADEN FORROW, AND JOHN R. SCHMITT Abstract. We present a restricted variable generalization of Warning s Second Theorem (a result giving

More information

WARNING S SECOND THEOREM WITH RESTRICTED VARIABLES

WARNING S SECOND THEOREM WITH RESTRICTED VARIABLES COMBINATORICA Bolyai Society Springer-Verlag Combinatorica 21pp. DOI: 10.1007/s00493-015-3267-8 WARNING S SECOND THEOREM WITH RESTRICTED VARIABLES PETE L. CLARK, ADEN FORROW, JOHN R. SCHMITT Received May

More information

On Zeros of a Polynomial in a Finite Grid: the Alon-Füredi Bound

On Zeros of a Polynomial in a Finite Grid: the Alon-Füredi Bound On Zeros of a Polynomial in a Finite Grid: the Alon-Füredi Bound John R. Schmitt Middlebury College, Vermont, USA joint work with Anurag Bishnoi (Ghent), Pete L. Clark (U. Georgia), Aditya Potukuchi (Rutgers)

More information

FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.

FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0

More information

Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille

Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is

More information

= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2

= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2 8. p-adic numbers 8.1. Motivation: Solving x 2 a (mod p n ). Take an odd prime p, and ( an) integer a coprime to p. Then, as we know, x 2 a (mod p) has a solution x Z iff = 1. In this case we can suppose

More information

x = π m (a 0 + a 1 π + a 2 π ) where a i R, a 0 = 0, m Z.

x = π m (a 0 + a 1 π + a 2 π ) where a i R, a 0 = 0, m Z. ALGEBRAIC NUMBER THEORY LECTURE 7 NOTES Material covered: Local fields, Hensel s lemma. Remark. The non-archimedean topology: Recall that if K is a field with a valuation, then it also is a metric space

More information

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)

More information

CHAPTER 14. Ideals and Factor Rings

CHAPTER 14. Ideals and Factor Rings CHAPTER 14 Ideals and Factor Rings Ideals Definition (Ideal). A subring A of a ring R is called a (two-sided) ideal of R if for every r 2 R and every a 2 A, ra 2 A and ar 2 A. Note. (1) A absorbs elements

More information

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of

More information

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................

More information

Factorization in Polynomial Rings

Factorization in Polynomial Rings Factorization in Polynomial Rings Throughout these notes, F denotes a field. 1 Long division with remainder We begin with some basic definitions. Definition 1.1. Let f, g F [x]. We say that f divides g,

More information

CHAPTER I. Rings. Definition A ring R is a set with two binary operations, addition + and

CHAPTER I. Rings. Definition A ring R is a set with two binary operations, addition + and CHAPTER I Rings 1.1 Definitions and Examples Definition 1.1.1. A ring R is a set with two binary operations, addition + and multiplication satisfying the following conditions for all a, b, c in R : (i)

More information

Mathematics for Cryptography

Mathematics for Cryptography Mathematics for Cryptography Douglas R. Stinson David R. Cheriton School of Computer Science University of Waterloo Waterloo, Ontario, N2L 3G1, Canada March 15, 2016 1 Groups and Modular Arithmetic 1.1

More information

K. Johnson Department of Mathematics, Dalhousie University, Halifax, Nova Scotia, Canada D. Patterson

K. Johnson Department of Mathematics, Dalhousie University, Halifax, Nova Scotia, Canada D. Patterson #A21 INTEGERS 11 (2011) PROJECTIVE P -ORDERINGS AND HOMOGENEOUS INTEGER-VALUED POLYNOMIALS K. Johnson Department of Mathematics, Dalhousie University, Halifax, Nova Scotia, Canada johnson@mathstat.dal.ca

More information

(Rgs) Rings Math 683L (Summer 2003)

(Rgs) Rings Math 683L (Summer 2003) (Rgs) Rings Math 683L (Summer 2003) We will first summarise the general results that we will need from the theory of rings. A unital ring, R, is a set equipped with two binary operations + and such that

More information

MATH 326: RINGS AND MODULES STEFAN GILLE

MATH 326: RINGS AND MODULES STEFAN GILLE MATH 326: RINGS AND MODULES STEFAN GILLE 1 2 STEFAN GILLE 1. Rings We recall first the definition of a group. 1.1. Definition. Let G be a non empty set. The set G is called a group if there is a map called

More information

Math 210B. Artin Rees and completions

Math 210B. Artin Rees and completions Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an A-module. In class we defined the I-adic completion of M to be M = lim M/I n M. We will soon show

More information

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2018 57 5. p-adic Numbers 5.1. Motivating examples. We all know that 2 is irrational, so that 2 is not a square in the rational field Q, but that we can

More information

14 Ordinary and supersingular elliptic curves

14 Ordinary and supersingular elliptic curves 18.783 Elliptic Curves Spring 2015 Lecture #14 03/31/2015 14 Ordinary and supersingular elliptic curves Let E/k be an elliptic curve over a field of positive characteristic p. In Lecture 7 we proved that

More information

INTEGER VALUED POLYNOMIALS AND LUBIN-TATE FORMAL GROUPS

INTEGER VALUED POLYNOMIALS AND LUBIN-TATE FORMAL GROUPS INTEGER VALUED POLYNOMIALS AND LUBIN-TATE FORMAL GROUPS EHUD DE SHALIT AND ERAN ICELAND Abstract. If R is an integral domain and K is its field of fractions, we let Int(R) stand for the subring of K[x]

More information

Notes on Systems of Linear Congruences

Notes on Systems of Linear Congruences MATH 324 Summer 2012 Elementary Number Theory Notes on Systems of Linear Congruences In this note we will discuss systems of linear congruences where the moduli are all different. Definition. Given the

More information

Higher-Dimensional Analogues of the Combinatorial Nullstellensatz

Higher-Dimensional Analogues of the Combinatorial Nullstellensatz Higher-Dimensional Analogues of the Combinatorial Nullstellensatz Jake Mundo July 21, 2016 Abstract The celebrated Combinatorial Nullstellensatz of Alon describes the form of a polynomial which vanishes

More information

Chapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples

Chapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples Chapter 3 Rings Rings are additive abelian groups with a second operation called multiplication. The connection between the two operations is provided by the distributive law. Assuming the results of Chapter

More information

Integral Extensions. Chapter Integral Elements Definitions and Comments Lemma

Integral Extensions. Chapter Integral Elements Definitions and Comments Lemma Chapter 2 Integral Extensions 2.1 Integral Elements 2.1.1 Definitions and Comments Let R be a subring of the ring S, and let α S. We say that α is integral over R if α isarootofamonic polynomial with coefficients

More information

Math 4400, Spring 08, Sample problems Final Exam.

Math 4400, Spring 08, Sample problems Final Exam. Math 4400, Spring 08, Sample problems Final Exam. 1. Groups (1) (a) Let a be an element of a group G. Define the notions of exponent of a and period of a. (b) Suppose a has a finite period. Prove that

More information

LECTURE NOTES IN CRYPTOGRAPHY

LECTURE NOTES IN CRYPTOGRAPHY 1 LECTURE NOTES IN CRYPTOGRAPHY Thomas Johansson 2005/2006 c Thomas Johansson 2006 2 Chapter 1 Abstract algebra and Number theory Before we start the treatment of cryptography we need to review some basic

More information

Finite Fields. Sophie Huczynska. Semester 2, Academic Year

Finite Fields. Sophie Huczynska. Semester 2, Academic Year Finite Fields Sophie Huczynska Semester 2, Academic Year 2005-06 2 Chapter 1. Introduction Finite fields is a branch of mathematics which has come to the fore in the last 50 years due to its numerous applications,

More information

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12 MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

More information

MINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS

MINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS MINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS LORENZ HALBEISEN, MARTIN HAMILTON, AND PAVEL RŮŽIČKA Abstract. A subset X of a group (or a ring, or a field) is called generating, if the smallest subgroup

More information

Math 120 HW 9 Solutions

Math 120 HW 9 Solutions Math 120 HW 9 Solutions June 8, 2018 Question 1 Write down a ring homomorphism (no proof required) f from R = Z[ 11] = {a + b 11 a, b Z} to S = Z/35Z. The main difficulty is to find an element x Z/35Z

More information

Algebraic function fields

Algebraic function fields Algebraic function fields 1 Places Definition An algebraic function field F/K of one variable over K is an extension field F K such that F is a finite algebraic extension of K(x) for some element x F which

More information

Rings and groups. Ya. Sysak

Rings and groups. Ya. Sysak Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R -module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...

More information

Local Fields. Chapter Absolute Values and Discrete Valuations Definitions and Comments

Local Fields. Chapter Absolute Values and Discrete Valuations Definitions and Comments Chapter 9 Local Fields The definition of global field varies in the literature, but all definitions include our primary source of examples, number fields. The other fields that are of interest in algebraic

More information

Extension theorems for homomorphisms

Extension theorems for homomorphisms Algebraic Geometry Fall 2009 Extension theorems for homomorphisms In this note, we prove some extension theorems for homomorphisms from rings to algebraically closed fields. The prototype is the following

More information

MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM

MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM Basic Questions 1. Compute the factor group Z 3 Z 9 / (1, 6). The subgroup generated by (1, 6) is

More information

4. Noether normalisation

4. Noether normalisation 4. Noether normalisation We shall say that a ring R is an affine ring (or affine k-algebra) if R is isomorphic to a polynomial ring over a field k with finitely many indeterminates modulo an ideal, i.e.,

More information

PolynomialExponential Equations in Two Variables

PolynomialExponential Equations in Two Variables journal of number theory 62, 428438 (1997) article no. NT972058 PolynomialExponential Equations in Two Variables Scott D. Ahlgren* Department of Mathematics, University of Colorado, Boulder, Campus Box

More information

Homework 10 M 373K by Mark Lindberg (mal4549)

Homework 10 M 373K by Mark Lindberg (mal4549) Homework 10 M 373K by Mark Lindberg (mal4549) 1. Artin, Chapter 11, Exercise 1.1. Prove that 7 + 3 2 and 3 + 5 are algebraic numbers. To do this, we must provide a polynomial with integer coefficients

More information

Chapter 1 : The language of mathematics.

Chapter 1 : The language of mathematics. MAT 200, Logic, Language and Proof, Fall 2015 Summary Chapter 1 : The language of mathematics. Definition. A proposition is a sentence which is either true or false. Truth table for the connective or :

More information

RINGS: SUMMARY OF MATERIAL

RINGS: SUMMARY OF MATERIAL RINGS: SUMMARY OF MATERIAL BRIAN OSSERMAN This is a summary of terms used and main results proved in the subject of rings, from Chapters 11-13 of Artin. Definitions not included here may be considered

More information

MATH RING ISOMORPHISM THEOREMS

MATH RING ISOMORPHISM THEOREMS MATH 371 - RING ISOMORPHISM THEOREMS DR. ZACHARY SCHERR 1. Theory In this note we prove all four isomorphism theorems for rings, and provide several examples on how they get used to describe quotient rings.

More information

FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS

FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ

More information

Rings. Chapter 1. Definition 1.2. A commutative ring R is a ring in which multiplication is commutative. That is, ab = ba for all a, b R.

Rings. Chapter 1. Definition 1.2. A commutative ring R is a ring in which multiplication is commutative. That is, ab = ba for all a, b R. Chapter 1 Rings We have spent the term studying groups. A group is a set with a binary operation that satisfies certain properties. But many algebraic structures such as R, Z, and Z n come with two binary

More information

ZEROS OF SPARSE POLYNOMIALS OVER LOCAL FIELDS OF CHARACTERISTIC p

ZEROS OF SPARSE POLYNOMIALS OVER LOCAL FIELDS OF CHARACTERISTIC p ZEROS OF SPARSE POLYNOMIALS OVER LOCAL FIELDS OF CHARACTERISTIC p BJORN POONEN 1. Statement of results Let K be a field of characteristic p > 0 equipped with a valuation v : K G taking values in an ordered

More information

Formal Modules. Elliptic Modules Learning Seminar. Andrew O Desky. October 6, 2017

Formal Modules. Elliptic Modules Learning Seminar. Andrew O Desky. October 6, 2017 Formal Modules Elliptic Modules Learning Seminar Andrew O Desky October 6, 2017 In short, a formal module is to a commutative formal group as a module is to its underlying abelian group. For the purpose

More information

Class numbers of cubic cyclic. By Koji UCHIDA. (Received April 22, 1973)

Class numbers of cubic cyclic. By Koji UCHIDA. (Received April 22, 1973) J. Math. Vol. 26, Soc. Japan No. 3, 1974 Class numbers of cubic cyclic fields By Koji UCHIDA (Received April 22, 1973) Let n be any given positive integer. It is known that there exist real. (imaginary)

More information

6 Lecture 6: More constructions with Huber rings

6 Lecture 6: More constructions with Huber rings 6 Lecture 6: More constructions with Huber rings 6.1 Introduction Recall from Definition 5.2.4 that a Huber ring is a commutative topological ring A equipped with an open subring A 0, such that the subspace

More information

Places of Number Fields and Function Fields MATH 681, Spring 2018

Places of Number Fields and Function Fields MATH 681, Spring 2018 Places of Number Fields and Function Fields MATH 681, Spring 2018 From now on we will denote the field Z/pZ for a prime p more compactly by F p. More generally, for q a power of a prime p, F q will denote

More information

Lifting to non-integral idempotents

Lifting to non-integral idempotents Journal of Pure and Applied Algebra 162 (2001) 359 366 www.elsevier.com/locate/jpaa Lifting to non-integral idempotents Georey R. Robinson School of Mathematics and Statistics, University of Birmingham,

More information

RATIONAL LINEAR SPACES ON HYPERSURFACES OVER QUASI-ALGEBRAICALLY CLOSED FIELDS

RATIONAL LINEAR SPACES ON HYPERSURFACES OVER QUASI-ALGEBRAICALLY CLOSED FIELDS RATIONAL LINEAR SPACES ON HYPERSURFACES OVER QUASI-ALGEBRAICALLY CLOSED FIELDS TODD COCHRANE, CRAIG V. SPENCER, AND HEE-SUNG YANG Abstract. Let k = F q (t) be the rational function field over F q and f(x)

More information

Formal groups. Peter Bruin 2 March 2006

Formal groups. Peter Bruin 2 March 2006 Formal groups Peter Bruin 2 March 2006 0. Introduction The topic of formal groups becomes important when we want to deal with reduction of elliptic curves. Let R be a discrete valuation ring with field

More information

A few exercises. 1. Show that f(x) = x 4 x 2 +1 is irreducible in Q[x]. Find its irreducible factorization in

A few exercises. 1. Show that f(x) = x 4 x 2 +1 is irreducible in Q[x]. Find its irreducible factorization in A few exercises 1. Show that f(x) = x 4 x 2 +1 is irreducible in Q[x]. Find its irreducible factorization in F 2 [x]. solution. Since f(x) is a primitive polynomial in Z[x], by Gauss lemma it is enough

More information

Formal Groups. Niki Myrto Mavraki

Formal Groups. Niki Myrto Mavraki Formal Groups Niki Myrto Mavraki Contents 1. Introduction 1 2. Some preliminaries 2 3. Formal Groups (1 dimensional) 2 4. Groups associated to formal groups 9 5. The Invariant Differential 11 6. The Formal

More information

MATH 631: ALGEBRAIC GEOMETRY: HOMEWORK 1 SOLUTIONS

MATH 631: ALGEBRAIC GEOMETRY: HOMEWORK 1 SOLUTIONS MATH 63: ALGEBRAIC GEOMETRY: HOMEWORK SOLUTIONS Problem. (a.) The (t + ) (t + ) minors m (A),..., m k (A) of an n m matrix A are polynomials in the entries of A, and m i (A) = 0 for all i =,..., k if and

More information

EXERCISES. = {1, 4}, and. The zero coset is J. Thus, by (***), to say that J 4- a iu not zero, is to

EXERCISES. = {1, 4}, and. The zero coset is J. Thus, by (***), to say that J 4- a iu not zero, is to 19 CHAPTER NINETEEN Whenever J is a prime ideal of a commutative ring with unity A, the quotient ring A/J is an integral domain. (The details are left as an exercise.) An ideal of a ring is called proper

More information

Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35

Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime

More information

Factorization of integer-valued polynomials with square-free denominator

Factorization of integer-valued polynomials with square-free denominator accepted by Comm. Algebra (2013) Factorization of integer-valued polynomials with square-free denominator Giulio Peruginelli September 9, 2013 Dedicated to Marco Fontana on the occasion of his 65th birthday

More information

be any ring homomorphism and let s S be any element of S. Then there is a unique ring homomorphism

be any ring homomorphism and let s S be any element of S. Then there is a unique ring homomorphism 21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UFD. Therefore

More information

Formal power series rings, inverse limits, and I-adic completions of rings

Formal power series rings, inverse limits, and I-adic completions of rings Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely

More information

ON A THEOREM OF MORLAYE AND JOLY AND ITS GENERALIZATION arxiv: v2 [math.nt] 14 Dec 2017

ON A THEOREM OF MORLAYE AND JOLY AND ITS GENERALIZATION arxiv: v2 [math.nt] 14 Dec 2017 ON A THEOREM OF MORLAYE AND JOLY AND ITS GENERALIZATION arxiv:707.00353v2 [math.nt] 4 Dec 207 IOULIA N. BAOULINA Abstract. We show that a weaker version of the well-known theorem of Morlaye and Joly on

More information

9 - The Combinatorial Nullstellensatz

9 - The Combinatorial Nullstellensatz 9 - The Combinatorial Nullstellensatz Jacques Verstraëte jacques@ucsd.edu Hilbert s nullstellensatz says that if F is an algebraically closed field and f and g 1, g 2,..., g m are polynomials in F[x 1,

More information

Math 2070BC Term 2 Weeks 1 13 Lecture Notes

Math 2070BC Term 2 Weeks 1 13 Lecture Notes Math 2070BC 2017 18 Term 2 Weeks 1 13 Lecture Notes Keywords: group operation multiplication associative identity element inverse commutative abelian group Special Linear Group order infinite order cyclic

More information

NOTES ON FINITE FIELDS

NOTES ON FINITE FIELDS NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining

More information

10. Noether Normalization and Hilbert s Nullstellensatz

10. Noether Normalization and Hilbert s Nullstellensatz 10. Noether Normalization and Hilbert s Nullstellensatz 91 10. Noether Normalization and Hilbert s Nullstellensatz In the last chapter we have gained much understanding for integral and finite ring extensions.

More information

IRREDUCIBILITY TESTS IN Q[T ]

IRREDUCIBILITY TESTS IN Q[T ] IRREDUCIBILITY TESTS IN Q[T ] KEITH CONRAD 1. Introduction For a general field F there is no simple way to determine if an arbitrary polynomial in F [T ] is irreducible. Here we will focus on the case

More information

Some practice problems for midterm 2

Some practice problems for midterm 2 Some practice problems for midterm 2 Kiumars Kaveh November 14, 2011 Problem: Let Z = {a G ax = xa, x G} be the center of a group G. Prove that Z is a normal subgroup of G. Solution: First we prove Z is

More information

CDM. Finite Fields. Klaus Sutner Carnegie Mellon University. Fall 2018

CDM. Finite Fields. Klaus Sutner Carnegie Mellon University. Fall 2018 CDM Finite Fields Klaus Sutner Carnegie Mellon University Fall 2018 1 Ideals The Structure theorem Where Are We? 3 We know that every finite field carries two apparently separate structures: additive and

More information

Polynomials, Ideals, and Gröbner Bases

Polynomials, Ideals, and Gröbner Bases Polynomials, Ideals, and Gröbner Bases Notes by Bernd Sturmfels for the lecture on April 10, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra We fix a field K. Some examples of fields

More information

Prime Rational Functions and Integral Polynomials. Jesse Larone, Bachelor of Science. Mathematics and Statistics

Prime Rational Functions and Integral Polynomials. Jesse Larone, Bachelor of Science. Mathematics and Statistics Prime Rational Functions and Integral Polynomials Jesse Larone, Bachelor of Science Mathematics and Statistics Submitted in partial fulfillment of the requirements for the degree of Master of Science Faculty

More information

ON THE NUMBER OF SUBSEQUENCES WITH GIVEN SUM OF SEQUENCES IN FINITE ABELIAN p-groups

ON THE NUMBER OF SUBSEQUENCES WITH GIVEN SUM OF SEQUENCES IN FINITE ABELIAN p-groups ON THE NUMBER OF SUBSEQUENCES WITH GIVEN SUM OF SEQUENCES IN FINITE ABELIAN p-groups WEIDONG GAO AND ALFRED GEROLDINGER Abstract. Let G be an additive finite abelian p-group. For a given (long) sequence

More information

ϕ : Z F : ϕ(t) = t 1 =

ϕ : Z F : ϕ(t) = t 1 = 1. Finite Fields The first examples of finite fields are quotient fields of the ring of integers Z: let t > 1 and define Z /t = Z/(tZ) to be the ring of congruence classes of integers modulo t: in practical

More information

A CONSTRUCTION FOR ABSOLUTE VALUES IN POLYNOMIAL RINGS. than define a second approximation V 0

A CONSTRUCTION FOR ABSOLUTE VALUES IN POLYNOMIAL RINGS. than define a second approximation V 0 A CONSTRUCTION FOR ABSOLUTE VALUES IN POLYNOMIAL RINGS by SAUNDERS MacLANE 1. Introduction. An absolute value of a ring is a function b which has some of the formal properties of the ordinary absolute

More information

4.5 Hilbert s Nullstellensatz (Zeros Theorem)

4.5 Hilbert s Nullstellensatz (Zeros Theorem) 4.5 Hilbert s Nullstellensatz (Zeros Theorem) We develop a deep result of Hilbert s, relating solutions of polynomial equations to ideals of polynomial rings in many variables. Notation: Put A = F[x 1,...,x

More information

Moreover this binary operation satisfies the following properties

Moreover this binary operation satisfies the following properties Contents 1 Algebraic structures 1 1.1 Group........................................... 1 1.1.1 Definitions and examples............................. 1 1.1.2 Subgroup.....................................

More information

MATH 361: NUMBER THEORY FOURTH LECTURE

MATH 361: NUMBER THEORY FOURTH LECTURE MATH 361: NUMBER THEORY FOURTH LECTURE 1. Introduction Everybody knows that three hours after 10:00, the time is 1:00. That is, everybody is familiar with modular arithmetic, the usual arithmetic of the

More information

2 ALGEBRA II. Contents

2 ALGEBRA II. Contents ALGEBRA II 1 2 ALGEBRA II Contents 1. Results from elementary number theory 3 2. Groups 4 2.1. Denition, Subgroup, Order of an element 4 2.2. Equivalence relation, Lagrange's theorem, Cyclic group 9 2.3.

More information

where c R and the content of f is one. 1

where c R and the content of f is one. 1 9. Gauss Lemma Obviously it would be nice to have some more general methods of proving that a given polynomial is irreducible. The first is rather beautiful and due to Gauss. The basic idea is as follows.

More information

August 2015 Qualifying Examination Solutions

August 2015 Qualifying Examination Solutions August 2015 Qualifying Examination Solutions If you have any difficulty with the wording of the following problems please contact the supervisor immediately. All persons responsible for these problems,

More information

Ideals, congruence modulo ideal, factor rings

Ideals, congruence modulo ideal, factor rings Ideals, congruence modulo ideal, factor rings Sergei Silvestrov Spring term 2011, Lecture 6 Contents of the lecture Homomorphisms of rings Ideals Factor rings Typeset by FoilTEX Congruence in F[x] and

More information

Review of Linear Algebra

Review of Linear Algebra Review of Linear Algebra Throughout these notes, F denotes a field (often called the scalars in this context). 1 Definition of a vector space Definition 1.1. A F -vector space or simply a vector space

More information

Computations/Applications

Computations/Applications Computations/Applications 1. Find the inverse of x + 1 in the ring F 5 [x]/(x 3 1). Solution: We use the Euclidean Algorithm: x 3 1 (x + 1)(x + 4x + 1) + 3 (x + 1) 3(x + ) + 0. Thus 3 (x 3 1) + (x + 1)(4x

More information

4.4 Noetherian Rings

4.4 Noetherian Rings 4.4 Noetherian Rings Recall that a ring A is Noetherian if it satisfies the following three equivalent conditions: (1) Every nonempty set of ideals of A has a maximal element (the maximal condition); (2)

More information

8 Complete fields and valuation rings

8 Complete fields and valuation rings 18.785 Number theory I Fall 2017 Lecture #8 10/02/2017 8 Complete fields and valuation rings In order to make further progress in our investigation of finite extensions L/K of the fraction field K of a

More information

PROBLEMS ON CONGRUENCES AND DIVISIBILITY

PROBLEMS ON CONGRUENCES AND DIVISIBILITY PROBLEMS ON CONGRUENCES AND DIVISIBILITY 1. Do there exist 1,000,000 consecutive integers each of which contains a repeated prime factor? 2. A positive integer n is powerful if for every prime p dividing

More information

On Permutation Polynomials over Local Finite Commutative Rings

On Permutation Polynomials over Local Finite Commutative Rings International Journal of Algebra, Vol. 12, 2018, no. 7, 285-295 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2018.8935 On Permutation Polynomials over Local Finite Commutative Rings Javier

More information

Factorization in Integral Domains II

Factorization in Integral Domains II Factorization in Integral Domains II 1 Statement of the main theorem Throughout these notes, unless otherwise specified, R is a UFD with field of quotients F. The main examples will be R = Z, F = Q, and

More information

ALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes!

ALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes! ALGEBRAIC GROUPS Disclaimer: There are millions of errors in these notes! 1. Some algebraic geometry The subject of algebraic groups depends on the interaction between algebraic geometry and group theory.

More information

David Adam. Jean-Luc Chabert LAMFA CNRS-UMR 6140, Université de Picardie, France

David Adam. Jean-Luc Chabert LAMFA CNRS-UMR 6140, Université de Picardie, France #A37 INTEGERS 10 (2010), 437-451 SUBSETS OF Z WITH SIMULTANEOUS ORDERINGS David Adam GAATI, Université de la Polynésie Française, Tahiti, Polynésie Française david.adam@upf.pf Jean-Luc Chabert LAMFA CNRS-UMR

More information

CHAPTER 10: POLYNOMIALS (DRAFT)

CHAPTER 10: POLYNOMIALS (DRAFT) CHAPTER 10: POLYNOMIALS (DRAFT) LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN The material in this chapter is fairly informal. Unlike earlier chapters, no attempt is made to rigorously

More information

THUE S LEMMA AND IDONEAL QUADRATIC FORMS

THUE S LEMMA AND IDONEAL QUADRATIC FORMS THUE S LEMMA AND IDONEAL QUADRATIC FORMS PETE L. CLARK Abstract. The representation of integers by binary quadratic forms is one of the oldest problems in number theory, going back at least to Fermat s

More information

TOTALLY RAMIFIED PRIMES AND EISENSTEIN POLYNOMIALS. 1. Introduction

TOTALLY RAMIFIED PRIMES AND EISENSTEIN POLYNOMIALS. 1. Introduction TOTALLY RAMIFIED PRIMES AND EISENSTEIN POLYNOMIALS KEITH CONRAD A (monic) polynomial in Z[T ], 1. Introduction f(t ) = T n + c n 1 T n 1 + + c 1 T + c 0, is Eisenstein at a prime p when each coefficient

More information

CHAPTER 1. AFFINE ALGEBRAIC VARIETIES

CHAPTER 1. AFFINE ALGEBRAIC VARIETIES CHAPTER 1. AFFINE ALGEBRAIC VARIETIES During this first part of the course, we will establish a correspondence between various geometric notions and algebraic ones. Some references for this part of the

More information

GALOIS GROUPS AS PERMUTATION GROUPS

GALOIS GROUPS AS PERMUTATION GROUPS GALOIS GROUPS AS PERMUTATION GROUPS KEITH CONRAD 1. Introduction A Galois group is a group of field automorphisms under composition. By looking at the effect of a Galois group on field generators we can

More information

Algebra Exam Topics. Updated August 2017

Algebra Exam Topics. Updated August 2017 Algebra Exam Topics Updated August 2017 Starting Fall 2017, the Masters Algebra Exam will have 14 questions. Of these students will answer the first 8 questions from Topics 1, 2, and 3. They then have

More information

TOTALLY RAMIFIED PRIMES AND EISENSTEIN POLYNOMIALS. 1. Introduction

TOTALLY RAMIFIED PRIMES AND EISENSTEIN POLYNOMIALS. 1. Introduction TOTALLY RAMIFIED PRIMES AND EISENSTEIN POLYNOMIALS KEITH CONRAD A (monic) polynomial in Z[T ], 1. Introduction f(t ) = T n + c n 1 T n 1 + + c 1 T + c 0, is Eisenstein at a prime p when each coefficient

More information

Math 203A - Solution Set 3

Math 203A - Solution Set 3 Math 03A - Solution Set 3 Problem 1 Which of the following algebraic sets are isomorphic: (i) A 1 (ii) Z(xy) A (iii) Z(x + y ) A (iv) Z(x y 5 ) A (v) Z(y x, z x 3 ) A Answer: We claim that (i) and (v)

More information

φ(xy) = (xy) n = x n y n = φ(x)φ(y)

φ(xy) = (xy) n = x n y n = φ(x)φ(y) Groups 1. (Algebra Comp S03) Let A, B and C be normal subgroups of a group G with A B. If A C = B C and AC = BC then prove that A = B. Let b B. Since b = b1 BC = AC, there are a A and c C such that b =

More information

A finite universal SAGBI basis for the kernel of a derivation. Osaka Journal of Mathematics. 41(4) P.759-P.792

A finite universal SAGBI basis for the kernel of a derivation. Osaka Journal of Mathematics. 41(4) P.759-P.792 Title Author(s) A finite universal SAGBI basis for the kernel of a derivation Kuroda, Shigeru Citation Osaka Journal of Mathematics. 4(4) P.759-P.792 Issue Date 2004-2 Text Version publisher URL https://doi.org/0.890/838

More information