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1 Quadratic extensions Definition: Let R, S be commutative rings, R S. An extension of rings R S is said to be quadratic there is α S \R and monic polynomial f(x) R[x] of degree such that f(α) = 0 and S = R[α]. A domain R is said to be integrally closed if R is integrally closed in its quotient field Q. (1) Let R S be quadratic extension with α S \ R and monic polynomial f(x) R[x] of degree such that f(α) = 0 and S = R[α]. Show that: (i) R[α] = {a + bα a,b R}. (ii) there is α S such that f(x) = (x α)(x α) and R[α] = R[α]. () Let R S be quadratic extension of domains with α S \ R and monic polynomial f(x) R[x] of degree such that f(α) = 0 and S = R[α]. Let R be integrally closed domain. Show that: (i) For every s S \ R there is uniquely determined s S such that (x s)(x s) R[x]. Moreover, for s = a + bα, a,b R, holds s = a + bα, where α is the (uniquely determined) second root of f. (ii) For every x S there are uniquely determined a,b R such that x = a + bα. (iii) For a R put a = a. The map : S S, s s is a ring automorphism. This automorphism is the only non-identical automorphism of S as R-algebra. (iv) The map N : S R defined by N(x) = xx (called a multiplicative norm) is well defined and multiplicative, i.e. N(xy) = N(x)N(y). (v) x S is invertible (in S) if and only if N(x) is invertible (in R). (vi) If N(x) is irreducible element in R, then x is irreducible element in S. (3) Prove that every unique factorization domain is integrally closed. (Hence Z is integrally closed.) (4) Let R = Z[i] (the ring of Gaussian integers). (i) Prove that R is a Euclidean domain with degree function N(a+ib) = a +b. (ii) Show that x R is a prime element in R if and only if either (a) x = a + bi, where a 0 b and N(x) = a + b is a prime number (e.g =,1 + = 5, + 3 = 13,...) or (b) x {±p, ±ip}, where p is a prime number such that p a + b for any a,b Z (e.g. 3,7,11,...). (iii) Let I = Rx be maximal ideal ( x is prime in R). Prove that (a) R/I = Z p, if N(x) = p P, and (b) R/I = F p (a field with p elements), if N(x) = p, where p P. (iv) Use the Euclidean algorithm to find the greatest common divisor of i and 10 5i in the ring Z[i]. (You should also remember how to express this GCD as a linear combination of the numbers you started with by working backwards 1

2 through the algorithm.) (v) Express 7 + 6i as a product of primes in Z[i]. (vi) Prove that R is a normal domain (i.e. for every prime ideal p of R the localisation R p is an integrally closed domain). Deduce that Z[i] is a Dedekind domain. Is Z[i] a PID? Justify your answer. (5) Let R = Z[ ]. (a) Show that R is a Euclidean domain under the usual complex norm. (b) Show that the only positive integer x for which x+ is a cube in R is 5. (c) Show that is an irreducible element of R. (d) Show that if x and y are positive integer solutions of x + = y 3, then x + and x are relatively prime. (e) Deduce that the only solution in positive integers of x + = y 3 is x = 5,y = 3. (6) Let R = Z[ 3] = {a + b 3 a,b Z}. (a) Why is R an integral domain? (b) What are the units in R? (c) Is the element irreducible in R? (d) If x,y R, and divides xy, does it follow that divides either x or y? Justify your answer. (7) Let R be the ring Z[ 5]. (a) Show that R is not a UFD. (Hint. Prove that the two factorizations of 6 given by ()(3) and (1+ 5)(1 5) are indeed distinct, that is, the factors are not associates of each other.) (b) Also show that the principal ideal (6) admits the factorization (6) = (,1 + 5)(,1 5)(3,1 + 5)(3,1 5) and that the ideals on the right are distinct prime ideals. (8) Let R = Z[ 6]. (i) Show that 10 = 5 and 10 = ( + 6)( 6) are two factorizations of 10 Z[ 6] that are essentially distinct in the sense that the factorizations cannot be transformed into one another by multiplying the factors by units. (ii) Find the factorizations of the principal ideals (), (5), ( + 6) and ( 6) into a product of prime ideals of R = Z[ 6]. (9) Let R = Z[ 10]. (a) Show that R is not a PID. (Hint: Show that 10 admits two essentially different factorizations into irreducible elements of R.) (b) Let P = 7, Show that R/P is isomorphic to Z 7. (10) (i) Determine the units in the rings Z[i] and Z[ 10]. If n is a squarefree

3 positive integer, then determine the units in the ring Z[ n]. (ii) Prove that is irreducible Z[ 10]. Then use the observation that divides 6 = (4 + 10(4 10) to prove that is not a prime in Z[ 10]. Deduce that Z[ 10] is not a PID (hence it is not an ED for any choice of degree function). (iii) Is Z[ 10] a UFD? (iv) What is wrong with the following argument? The ring homomorphism ev 10 : Z[X] C, f(x) f( 10) ( evaluation at 10 ) has image Z[ 10]. Since Z[X] is a Euclidean domain it is a PID. Quotients of PIDs are PIDs. Hence Z[ 10] is a PID. (11) Let R = Z[ d] where d 0,1 is a square-free element of Z. (i) Assume d < 1. Find R. (ii) Assume d = 1. Find R and its group structure. (1) Let d 3 mod 4 be a square-free positive integer and let R = Z[ 1+ d ] be the usual quadratic domain. Use N : R N 0 to denote the norm map. (a) If d > 3, show that the only units in R are ±1. (b) If a,b R with b 0, show that there exist q,r R with a = bq + r and N(r) (4 + d)n(b)/16. (Hint. If a/b = u + v d with u,v Q, then v is distance at most 1/4 from an integer or half integer.) (c) Conclude that R is a Euclidean domain for d = 3,7,11. (d) Show that R is not a u.f.d. when d = 15 or 3. (Hint. Emulate the proof you know for Z[ 5].) Let a be an element of a domain S with a 0 and a not a unit. Then a is said to be a pseudo-unit if given any b S we have either a b or a (b + u) for some unit u S. (e) Find all pseudo-units in Z. (f) If S is a Euclidean domain and not a field, show that S has a pseudo-unit. (Hint. First check that the set A of elements of S that are not zero and not units is nonempty. If S is Euclidean with ν : S \ 0 Z +, consider an element a A with ν(a) minimal.) (g) Let d = 19. Show that R has no pseudo-units. Hence R is not Euclidean. (Hint. Suppose a is a pseudo-unit. Using b =, conclude that N(a) divides 1, 4 or 9. Using b = 1+ 19, conclude that N(a) divides 5 or 7.) (Another hint. Show that the only elements of norm less than 5 are {0, ±1, ±}. Taking x = in the definition of pseudo-unit, show that any pseudo-unit must be a nonunit divisor of of 3. Find all nonunit divisors of and 3. Now taking x = in the definition of pseudo-unit, find a contradiction.) (h) Repeat the argument for d = 163. (13) Note that ω = 1+ 3 is a cube root of 1, and ω + ω + 1 = 0. (i) Prove that the subset {x + yω Z[ω] x + y is divisible by 3} is an ideal of Z[ω]. Is it prime? (ii) Show that the ring R = Z[ω] (sometimes called the Eisenstein integers) is a Euclidean domain with respect to the norm function N(z) = z z, (where, as 3

4 usual, z denotes the complex conjugate of z). Hence R is a Unique Factorization Domain. (Hint: recall the proof for the Gaussian integers Z[i].) (iii) Factor, 3, 5, and 7 into primes in R. (Which one of them has a repeated prime factor? This prime factor is key to an elementary proof of Fermat s Last Theorem for n = 3.) (iv) Prove that Z[ω]/(p) = Z p [x]/(x + x + 1) and, as a consequence, that a prime p in Z is expressible as x +xy +y with integers x,y if and only if p 1 mod 3. (14) Let τ = 1+ 5 be the golden mean. Show that Z[τ] has infinitely many units. Definition: Let Q F be field extension of finite degree (i.e. dim Q F < ). Denote Z F the integral closure of Z in F. F is called number field and Z F is called the ring of integers of number field F. For [F : Q] := dim Q F = is F called quadratic number field. (15) Let 1 d Z be square-free integer and F = Q( d). (a) If d,3 mod 4 then (O F =)Z F = Z[ d] and {1, d} is an integral basis. (b) If d 1 mod 4 then (O F =)Z F = Z[ 1+ d ] = Z[ 1+ d ] and {1, 1+ d } is an integral basis. (16) Let d be a squarefree integer, and let O = Z[ω] be the ring of integers in the quadratic field F = Q( d), where ω = 1+ d if D 1 (mod 4), and ω = d otherwise. (i) For any positive integer f prove that the set O f = Z[fω] = {a + bfω a,b Z} is a subring of O. (ii) Prove that [O : O f ] = f (where the index is as additive abelian groups). (iii) Prove conversely that a subring of O containing the identity and having finite index f in O is equal to O f. (The ring O f is called the order of conductor f in the field Q( d). The ring of integers O is called the maximal order in Q( d).) (17) A quadratic number field is a number field K with [K : Q] =. (a) Show that every quadratic number field K can be represented as K = Q( d) for a square-free integer d 0,1. (b) Let K := Q( d) for some square-free integer d 0,1. Show that: D K = { Z[ 1+ d ] = Z[x]/(x x + 1 d 4 ),if d 1 (mod 4) Z[ d] = Z[x]/(x d),if d,3 (mod 4) (In particular, the ring of integers of Q(i) = {a + ib a,b Q} is the ring Z[i] = {a + ib a,b Z} of Gaussian integers.) (18) Let F = Q( 5). Let I be ideal in the O F generated by and Show that D = (I,1/ ) is a Hermitian line bundle of degree 0. Draw pictures 4

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