ALGEBRAIC NUMBER THEORY PART II (SOLUTIONS) =
|
|
- Camilla Waters
- 5 years ago
- Views:
Transcription
1 ALGEBRAIC NUMBER THEORY PART II SOLUTIONS) 1. Eisenstein Integers Exercise 1. Let ω = Verify that ω + ω + 1 = 0. Solution. We have ) = = 1 1 ) ) = 0. Exercise. The set Z[ω] = {a + bω : a, b Z} is called the ring of Eisenstein integers. Prove that it is a ring. That is, show that for any Eisenstein integers a + bω, c + dω the numbers a + bω) + c + dω), a + bω) c + dω), a + bω)c + dω) are Eisenstein integers as well. Solution. For the sum and difference the calculations are easy: a + bω) + c + dω) = a + c) + b + d)ω, a + bω) c + dω) = a c) + b d)ω. Since a, b, c, d are integers, we see that a + c, b + d, a c, b d are integers as well. Therefore the above numbers are Eisenstein integers. The product is a bit more tricky. Here we need to use the fact that ω = 1 ω: a + bω)c + dω) = ac + adω + bcω + bdω = ac + adω + bcω + bd 1 ω) = ac bd) + ad + bc bd)ω. Since a, b, c, d are integers, we see that ac bd and ad + bc bd are integers as well, so the above number is an Einstein integer. Exercise 3. For any Eisenstein integer a + bω define the norm Na + bω) = a ab + b. Prove that the norm is multiplicative. That is, for any Eisenstein integers a + bω, c + dω the equality N a + bω)c + dω)) = Na + bω)nc + dω) 1
2 ALGEBRAIC NUMBER THEORY PART II SOLUTIONS) holds. Verify that the norm is non-negative: Na + bω) 0 for any a, b and Na + bω) = 0 if and only if a = b = 0. Solution. We have Na + bω) = a ab + b, Nc + dω) = c cd + d. Also, from the previous exercise we know that N a + bω)c + dω)) = N ac bd) + ad + bc bd)ω) = ac bd) ac bd)ad + bc bd) + ad + bc bd) = a ab + b )c cd + d ) = Na + bω)nc + dω). To see that the norm is non-negative, we need to show that Na + bω) = a ab + b 0 for all a, b. This becomes evident once we try to use the trick called completing the square : a ab + b = a a b + b = a a b + b + ) b = a a b + b ) + b = a b ) 4 b. b b Now it suddenly becomes very clear that this number is always non-negative, because it is a sum of two non-negative quantities a b/) and 3b /4. To see for which a, b it is possible to have Na + bω) = 0, we just need to solve the equation a b ) 4 b = 0 in integers a and b. If b 1, then clearly the left hand side will be greater than zero, so it must be the case that b = 0. Therefore a b/) = a = 0, which means that a = 0. Exercise 3. We say that γ is an Eisenstein unit if γ 1. Prove that if γ is an Eisenstein unit then its norm is equal to one. Solution. Suppose that γ is a unit, so γ 1. By definition, there exists an Eisenstein integer β such that βγ = 1. Applying the norm function on both sides of this equation, we get Nβγ) = N1). Then we use the multiplicative property of the norm to conclude that Nβ)Nγ) = 1. Since Nγ) is an integer, it has to be equal to either +1 or 1. However, in the previous exercise we established that the norm is non-negative, so Nγ) = 1. Conversely, suppose that Nγ) = 1. Write γ = a + bω. In order to show that γ is a unit we need to prove that γ 1, i.e. there exists some Eisenstein integer c + dω such that a + bω)c + dω) = 1. ) )
3 ALGEBRAIC NUMBER THEORY PART II SOLUTIONS) 3 Exercise 4. Find all Eisenstein integers of norm one there are six of them) and show that all of them are Eisenstein units. Solution. Let us find all integers a, b such that Na+bω) = 1. This is equivalent to solving the equation a ab + b = 1 in integers. We recall that this equation is the same as a b ) 4 b = 1. Clearly, if b, then the left hand side is at least 3, so it must be the case that b 1. Thus we need to check if integer solutions exist for b = 1, b = 0 or b = 1. In fact, to each of those b s correspond two values of a. The complete list of solutions is: a, b) = 1, 1), 1, 0), 0, 1), 0, 1), 1, 0), 1, 1). They correspond to Eisenstein integers 1 ω, 1, ω, ω, 1, 1 + ω. So far, we produced a complete list of Eisenstein integers that have norm one. Now we will prove that they are all units. For 1 and 1 this is obviously the case. We have 1 = ω 3 = ω ω = ω 1 ω). Since 1 ω is an Eisenstein integer, we see that ω 1, so it is a unit. From the above equation we can also see that 1 ω 1, because ω is an Eisenstein integer. Finally, we can also write the above equality as 1 = ω)1 + ω), which means that ω 1 and 1 + ω 1. Therefore all six Eisenstein integers of norm one that we have found are units. Exercise 5. An Eisenstein integer γ is prime if it is not a unit and every factorization γ = αβ with α, β Z[ω] forces one of α or β to be a unit. Find Eisenstein primes among rational primes, 3, 5, 7, 11, 13. Solution. Suppose that = a + bω)c + dω) for some Eisenstein integers a + bω, c + dω. In order to prove that is prime we need show that either a + bω or c + dω is a unit. Write 4 = N) = N a + bω)c + dω)) = Na + bω)nc + dω). Since Na + bω) is an integer dividing 4, it must be equal to either 1, or 4. If it is equal to 1 then the previous exercise tells us that it is a unit. If it is equal to 4 then the norm of c + dω is equal to 1, so it is a unit. Thus it remains to check the case Na + bω) =. For this purpose we need to solve the equation a ab + b = in integers a and b. This equation is equivalent to a b ) 4 b =. If b then the left hand side is at least 3, so it must be the case that b 1. We can easily verify that there are no integer a s that correspond to b = 1, 0 or 1. Therefore Na + bω), which means that Na + bω) = 1 or 4, so either a + bω
4 4 ALGEBRAIC NUMBER THEORY PART II SOLUTIONS) or c + dω is a unit. We conclude that is a Gaussian prime. Analogously, we can find out that 5 and 11 are Gaussian primes. To see that 3 is not a Gaussian prime, we need to solve the equation a b ) 4 b = 3. This equation has six solutions, one of which is a, b) = 1, 1), which corresponds to the Eisenstein integer 1 ω. We can now find c+dω such that 1 ω)c+dω) = 3. We have c =, d = 1, so 1 ω) + ω) = 3. We see that neither 1 ω nor + ω have norm equal to 1, which means that 3 is not an Eisenstein prime. Analogously, we can find integer solutions to equations a ab + b = 7, a ab + b = 13 and prove that neither 7 nor 13 are primes by factoring them into Eisenstein integers that are not units: 7 = 1 ω)3 + ω), 13 = 1 3ω)4 ω). Remark 1.1. As an exercise, try proving that every prime number of the form 3k +, like, 5, 11 or 17, is not an Eisenstein prime. This can be done by showing that every integer of the form a ab + b has to be either of the form 3k or of the form 3k + 1, but it can never take the form 3k +. Remark 1.. In fact, a bit more can be said about the rational prime 3: we can factor it as 3 = 1 + ω)1 ω), where 1 + ω is a unit. Therefore 3 is divisible by a square of an Eisenstein prime 1 ω. This means that 3 is a ramified prime. In fact, it is the only rational prime that ramifies.. Failure of Unique Factorization Exercise 1. Consider the ring Z[ 5] = { a + b 5: a, b Z } along with the norm map Na + b 5) = a + 5b, which is known to be multiplicative. Prove that ±1 are the only units in Z[ 5]. Solution. Suppose that a + b 5 is a unit, so it divides 1. Therefore a + b 5)c + d 5) = 1 for some integers c and d. Applying the norm function on both sides and then using its multiplicative property, we see that Na + b 5)Nc + d 5) = 1. Since Na + b 5) is an integer, it must be the case that Na + b 5) = a + 5b is equal to either 1 or 1. Clearly it cannot be negative, so we need to solve the equation a + 5b = 1 in integers a and b. If b 1, then the left hand side is at least 5, so it must be the case that b = 0. Therefore a = 1, which means that a = ±1.
5 ALGEBRAIC NUMBER THEORY PART II SOLUTIONS) 5 Exercise. Prove that the numbers, 3, 1+ 5, 1 5 are prime in Z[ 5]. Solution. Write = a + b 5)c + d 5). Then 4 = N) = N a + b 5)c + d 5) ) = Na + b 5)Nc + d 5). Therefore Na + b 5) is either equal to 1, or 4. If it is 1 then a + b 5 is a unit, while if it is 4 then c + d 5 is a unit. It remains to consider the case Na + b 5) =. However, it is quite easy to see that the equation a + 5b = has no solutions in integers a and b. Therefore is prime in Z[ 5]. Analogously we can prove this statement for 3 by proving that the equation Na + b 5) = 3 has no solutions in integers. For the logic is similar: we write = a + b 5)c + d 5) and observe that 6 = N1 + 5) = N a + b 5)c + d 5) ) = Na + b 5)Nc + d 5). Therefore Na + b 5) is equal to either 1,, 3 or 6. If it is 1 then a + b 5 is a unit and if it 6 then c + d 5 is a unit. Thus we need consider the cases Na + b 5) = and Na + b 5) = 3. However, they have already been considered, and we proved that they have no solutions. We conclude that is prime and analogously we can prove that 1 5 is prime. Exercise 3. Using Exercise prove that the unique factorization fails in Z[ 5]. Solution. Note that 6 = 3 = 1 + 5)1 5), so we obtain two factorizations of a number 6, so we obtained two different prime factorizations of a number 6 in Z[ 5]. 3. Rings with Infinitely Many Units Exercise 1. Find at least one unit different from ±1 in the rings Z[ ] and Z[ 5]. Hint: the norm map is Na+b d) = a db and it is multiplicative. Convince yourself that Na+b d) = ±1 and then find one solution to the Diophantine equation you obtained. Solution. Suppose that a + b is a unit, so a + b 1. Then there exist integers c and d such that Therefore a + b )c + d ) = 1. Na + b )Nc + d ) = 1. Since Na + b ) is an integer, we must have Na + b ) = ±1. Thus we need to solve the equation a b = ±1 in integers a and b. By observation, a = b = 1 is a solution. Analogously, if a + b 5 is a unit in Z[ 5], then Na + b 5) = ±1, so we need to solve the equation a 5b = ±1 in integers a and b. By observation, a =, b = 1 is a solution.
6 6 ALGEBRAIC NUMBER THEORY PART II SOLUTIONS) Exercise. Suppose that you have found a unit a + b d. Prove that for any positive integer n the number a + b d) n is also a unit. Prove that there are infinitely many units. Proof. For d = we have found a unit 1 +. Note that 1 + ) n 1 + ) n = 1 + ) 1 + n )) = 1. Since 1) n Z[ ] we see that 1 + ) n 1, so by definition it is a unit. To see that there are infinitely many units, just note that the sequence 1 + < 1 + ) < 1 + ) 3 <... is strictly increasing, so the numbers in it do not repeat. Analogously, we observe that + 5) n + 5) n = + 5) + n 5)) = 1. Since + 5) n Z[ 5], we conclude that + 5) n 1, so it is a unit. Further, + 5 < + 5) < + 5) 3 <... is a strictly increasing sequence, so there are infinitely many units.
Just like the ring of Gaussian integers, the ring of Eisenstein integers is a Unique Factorization Domain.
Fermat s Infinite Descent PMATH 340 Assignment 6 (Due Monday April 3rd at noon). (0 marks) Use Femtat s method of infinite descent to prove that the Diophantine equation x 3 + y 3 = 4z 3 has no solutions
More informationHomework due on Monday, October 22
Homework due on Monday, October 22 Read sections 2.3.1-2.3.3 in Cameron s book and sections 3.5-3.5.4 in Lauritzen s book. Solve the following problems: Problem 1. Consider the ring R = Z[ω] = {a+bω :
More informationGaussian integers. 1 = a 2 + b 2 = c 2 + d 2.
Gaussian integers 1 Units in Z[i] An element x = a + bi Z[i], a, b Z is a unit if there exists y = c + di Z[i] such that xy = 1. This implies 1 = x 2 y 2 = (a 2 + b 2 )(c 2 + d 2 ) But a 2, b 2, c 2, d
More informationAlgebraic number theory Solutions to exercise sheet for chapter 4
Algebraic number theory Solutions to exercise sheet for chapter 4 Nicolas Mascot n.a.v.mascot@warwick.ac.uk), Aurel Page a.r.page@warwick.ac.uk) TAs: Chris Birkbeck c.d.birkbeck@warwick.ac.uk), George
More informationFACTORING IN QUADRATIC FIELDS. 1. Introduction
FACTORING IN QUADRATIC FIELDS KEITH CONRAD For a squarefree integer d other than 1, let 1. Introduction K = Q[ d] = {x + y d : x, y Q}. This is closed under addition, subtraction, multiplication, and also
More information12. Quadratic Reciprocity in Number Fields
1. Quadratic Reciprocity in Number Fields In [RL1, p. 73/74] we briefly mentioned how Eisenstein arrived at a hypothetical reciprocity law by assuming what later turned out to be an immediate consequence
More information1. Factorization Divisibility in Z.
8 J. E. CREMONA 1.1. Divisibility in Z. 1. Factorization Definition 1.1.1. Let a, b Z. Then we say that a divides b and write a b if b = ac for some c Z: a b c Z : b = ac. Alternatively, we may say that
More informationDirect Proof Divisibility
Direct Proof Divisibility Lecture 15 Section 4.3 Robb T. Koether Hampden-Sydney College Fri, Feb 8, 2013 Robb T. Koether (Hampden-Sydney College) Direct Proof Divisibility Fri, Feb 8, 2013 1 / 20 1 Divisibility
More information18. Cyclotomic polynomials II
18. Cyclotomic polynomials II 18.1 Cyclotomic polynomials over Z 18.2 Worked examples Now that we have Gauss lemma in hand we can look at cyclotomic polynomials again, not as polynomials with coefficients
More informationIntroduction to Algebraic Number Theory Part I
Introduction to Algebraic Number Theory Part I A. S. Mosunov University of Waterloo Math Circles November 7th, 2018 Goals Explore the area of mathematics called Algebraic Number Theory. Specifically, we
More informationarxiv: v1 [math.nt] 29 Feb 2016
PERFECT NUMBERS IN THE RING OF EISENSTEIN INTEGERS ZACHARY PARKER, JEFF RUSHALL, AND JORDAN HUNT arxiv:1602.09106v1 [math.nt] 29 Feb 2016 Abstract. One of the many number theoretic topics investigated
More informationMath Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. A prime number
More information(January 14, 2009) q n 1 q d 1. D = q n = q + d
(January 14, 2009) [10.1] Prove that a finite division ring D (a not-necessarily commutative ring with 1 in which any non-zero element has a multiplicative inverse) is commutative. (This is due to Wedderburn.)
More informationMATH1050 Greatest/least element, upper/lower bound
MATH1050 Greatest/ element, upper/lower bound 1 Definition Let S be a subset of R x λ (a) Let λ S λ is said to be a element of S if, for any x S, x λ (b) S is said to have a element if there exists some
More information2.2 Some Consequences of the Completeness Axiom
60 CHAPTER 2. IMPORTANT PROPERTIES OF R 2.2 Some Consequences of the Completeness Axiom In this section, we use the fact that R is complete to establish some important results. First, we will prove that
More informationHomework 6 Solution. Math 113 Summer 2016.
Homework 6 Solution. Math 113 Summer 2016. 1. For each of the following ideals, say whether they are prime, maximal (hence also prime), or neither (a) (x 4 + 2x 2 + 1) C[x] (b) (x 5 + 24x 3 54x 2 + 6x
More informationRecall, R is an integral domain provided: R is a commutative ring If ab = 0 in R, then either a = 0 or b = 0.
Recall, R is an integral domain provided: R is a commutative ring If ab = 0 in R, then either a = 0 or b = 0. Examples: Z Q, R Polynomials over Z, Q, R, C The Gaussian Integers: Z[i] := {a + bi : a, b
More informationNOTES ON DIOPHANTINE APPROXIMATION
NOTES ON DIOPHANTINE APPROXIMATION Jan-Hendrik Evertse January 29, 200 9 p-adic Numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics
More informationTOTALLY 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 informationYes zero is a rational number as it can be represented in the
1 REAL NUMBERS EXERCISE 1.1 Q: 1 Is zero a rational number? Can you write it in the form 0?, where p and q are integers and q Yes zero is a rational number as it can be represented in the form, where p
More informationPRACTICE FINAL MATH , MIT, SPRING 13. You have three hours. This test is closed book, closed notes, no calculators.
PRACTICE FINAL MATH 18.703, MIT, SPRING 13 You have three hours. This test is closed book, closed notes, no calculators. There are 11 problems, and the total number of points is 180. Show all your work.
More informationALGEBRA PH.D. QUALIFYING EXAM SOLUTIONS October 20, 2011
ALGEBRA PH.D. QUALIFYING EXAM SOLUTIONS October 20, 2011 A passing paper consists of four problems solved completely plus significant progress on two other problems; moreover, the set of problems solved
More informationPrime Factorization and GCF. In my own words
Warm- up Problem What is a prime number? A PRIME number is an INTEGER greater than 1 with EXACTLY 2 positive factors, 1 and the number ITSELF. Examples of prime numbers: 2, 3, 5, 7 What is a composite
More informationMTH 310, Section 001 Abstract Algebra I and Number Theory. Sample Midterm 1
MTH 310, Section 001 Abstract Algebra I and Number Theory Sample Midterm 1 Instructions: You have 50 minutes to complete the exam. There are five problems, worth a total of fifty points. You may not use
More informationTHE P-ADIC NUMBERS AND FINITE FIELD EXTENSIONS OF Q p
THE P-ADIC NUMBERS AND FINITE FIELD EXTENSIONS OF Q p EVAN TURNER Abstract. This paper will focus on the p-adic numbers and their properties. First, we will examine the p-adic norm and look at some of
More informationChapter 8. P-adic numbers. 8.1 Absolute values
Chapter 8 P-adic numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.
More information1 The Well Ordering Principle, Induction, and Equivalence Relations
1 The Well Ordering Principle, Induction, and Equivalence Relations The set of natural numbers is the set N = f1; 2; 3; : : :g. (Some authors also include the number 0 in the natural numbers, but number
More informationTOTALLY 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 informationAlgebraic number theory Revision exercises
Algebraic number theory Revision exercises Nicolas Mascot (n.a.v.mascot@warwick.ac.uk) Aurel Page (a.r.page@warwick.ac.uk) TA: Pedro Lemos (lemos.pj@gmail.com) Version: March 2, 20 Exercise. What is the
More information6]. (10) (i) Determine the units in the rings Z[i] and Z[ 10]. If n is a squarefree
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
More informationSchool of Mathematics
School of Mathematics Programmes in the School of Mathematics Programmes including Mathematics Final Examination Final Examination 06 22498 MSM3P05 Level H Number Theory 06 16214 MSM4P05 Level M Number
More informationChapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
More informationThe Real Number System
MATH 337 The Real Number System Sets of Numbers Dr. Neal, WKU A set S is a well-defined collection of objects, with well-defined meaning that there is a specific description from which we can tell precisely
More informationTHE JOHNS HOPKINS UNIVERSITY Faculty of Arts and Sciences FINAL EXAM - FALL SESSION ADVANCED ALGEBRA I.
THE JOHNS HOPKINS UNIVERSITY Faculty of Arts and Sciences FINAL EXAM - FALL SESSION 2006 110.401 - ADVANCED ALGEBRA I. Examiner: Professor C. Consani Duration: take home final. No calculators allowed.
More informationIntegral Bases 1 / 14
Integral Bases 1 / 14 Overview Integral Bases Norms and Traces Existence of Integral Bases 2 / 14 Basis Let K = Q(θ) be an algebraic number field of degree n. By Theorem 6.5, we may assume without loss
More informationMATH 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 information32 Divisibility Theory in Integral Domains
3 Divisibility Theory in Integral Domains As we have already mentioned, the ring of integers is the prototype of integral domains. There is a divisibility relation on * : an integer b is said to be divisible
More informationRings, Integral Domains, and Fields
Rings, Integral Domains, and Fields S. F. Ellermeyer September 26, 2006 Suppose that A is a set of objects endowed with two binary operations called addition (and denoted by + ) and multiplication (denoted
More informationM3P14 LECTURE NOTES 8: QUADRATIC RINGS AND EUCLIDEAN DOMAINS
M3P14 LECTURE NOTES 8: QUADRATIC RINGS AND EUCLIDEAN DOMAINS 1. The Gaussian Integers Definition 1.1. The ring of Gaussian integers, denoted Z[i], is the subring of C consisting of all complex numbers
More informationMath Circle Beginners Group February 28, 2016 Euclid and Prime Numbers
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. (a) Circle the prime
More information1 First Theme: Sums of Squares
I will try to organize the work of this semester around several classical questions. The first is, When is a prime p the sum of two squares? The question was raised by Fermat who gave the correct answer
More informationThe p-adic numbers. Given a prime p, we define a valuation on the rationals by
The p-adic numbers There are quite a few reasons to be interested in the p-adic numbers Q p. They are useful for solving diophantine equations, using tools like Hensel s lemma and the Hasse principle,
More informationCHAPTER 1. REVIEW: NUMBERS
CHAPTER. REVIEW: NUMBERS Yes, mathematics deals with numbers. But doing math is not number crunching! Rather, it is a very complicated psychological process of learning and inventing. Just like listing
More information3 The language of proof
3 The language of proof After working through this section, you should be able to: (a) understand what is asserted by various types of mathematical statements, in particular implications and equivalences;
More informationProblems in Algebra. 2 4ac. 2a
Problems in Algebra Section Polynomial Equations For a polynomial equation P (x) = c 0 x n + c 1 x n 1 + + c n = 0, where c 0, c 1,, c n are complex numbers, we know from the Fundamental Theorem of Algebra
More informationALGEBRA AND NUMBER THEORY II: Solutions 3 (Michaelmas term 2008)
ALGEBRA AND NUMBER THEORY II: Solutions 3 Michaelmas term 28 A A C B B D 61 i If ϕ : R R is the indicated map, then ϕf + g = f + ga = fa + ga = ϕf + ϕg, and ϕfg = f ga = faga = ϕfϕg. ii Clearly g lies
More informationNOTES 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 information3.4 The Fundamental Theorem of Algebra
333371_0304.qxp 12/27/06 1:28 PM Page 291 3.4 The Fundamental Theorem of Algebra Section 3.4 The Fundamental Theorem of Algebra 291 The Fundamental Theorem of Algebra You know that an nth-degree polynomial
More informationDirect Proof Divisibility
Direct Proof Divisibility Lecture 15 Section 4.3 Robb T. Koether Hampden-Sydney College Fri, Feb 7, 2014 Robb T. Koether (Hampden-Sydney College) Direct Proof Divisibility Fri, Feb 7, 2014 1 / 23 1 Divisibility
More informationNumber Axioms. P. Danziger. A Group is a set S together with a binary operation (*) on S, denoted a b such that for all a, b. a b S.
Appendix A Number Axioms P. Danziger 1 Number Axioms 1.1 Groups Definition 1 A Group is a set S together with a binary operation (*) on S, denoted a b such that for all a, b and c S 0. (Closure) 1. (Associativity)
More informationHans Wenzl. 4f(x), 4x 3 + 4ax bx + 4c
MATH 104C NUMBER THEORY: NOTES Hans Wenzl 1. DUPLICATION FORMULA AND POINTS OF ORDER THREE We recall a number of useful formulas. If P i = (x i, y i ) are the points of intersection of a line with the
More informationCitation for published version (APA): Ruíz Duarte, E. An invitation to algebraic number theory and class field theory
University of Groningen An invitation to algebraic number theory and class field theory Ruíz Duarte, Eduardo IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you
More informationMath Lecture 18 Notes
Math 1010 - Lecture 18 Notes Dylan Zwick Fall 2009 In our last lecture we talked about how we can add, subtract, and multiply polynomials, and we figured out that, basically, if you can add, subtract,
More informationContinuing the pre/review of the simple (!?) case...
Continuing the pre/review of the simple (!?) case... Garrett 09-16-011 1 So far, we have sketched the connection between prime numbers, and zeros of the zeta function, given by Riemann s formula p m
More informationClass VIII Chapter 1 Rational Numbers Maths. Exercise 1.1
Question 1: Using appropriate properties find: Exercise 1.1 (By commutativity) Page 1 of 11 Question 2: Write the additive inverse of each of the following: (iii) (iv) (v) Additive inverse = Additive inverse
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 3 Due Friday, February 5 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 3 Due Friday, February 5 at 08:35 1. Let R 0 be a commutative ring with 1 and let S R be the subset of nonzero elements which are not zero divisors. (a)
More informationP-adic numbers. Rich Schwartz. October 24, 2014
P-adic numbers Rich Schwartz October 24, 2014 1 The Arithmetic of Remainders In class we have talked a fair amount about doing arithmetic with remainders and now I m going to explain what it means in a
More informationScott Taylor 1. EQUIVALENCE RELATIONS. Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that:
Equivalence MA Relations 274 and Partitions Scott Taylor 1. EQUIVALENCE RELATIONS Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that: (1) is reflexive. That is, (2) is
More informationARITHMETIC IN PURE CUBIC FIELDS AFTER DEDEKIND.
ARITHMETIC IN PURE CUBIC FIELDS AFTER DEDEKIND. IAN KIMING We will study the rings of integers and the decomposition of primes in cubic number fields K of type K = Q( 3 d) where d Z. Cubic number fields
More informationMath 504, Fall 2013 HW 2
Math 504, Fall 203 HW 2. Show that the fields Q( 5) and Q( 7) are not isomorphic. Suppose ϕ : Q( 5) Q( 7) is a field isomorphism. Then it s easy to see that ϕ fixes Q pointwise, so 5 = ϕ(5) = ϕ( 5 5) =
More informationMODEL ANSWERS TO HWK #10
MODEL ANSWERS TO HWK #10 1. (i) As x + 4 has degree one, either it divides x 3 6x + 7 or these two polynomials are coprime. But if x + 4 divides x 3 6x + 7 then x = 4 is a root of x 3 6x + 7, which it
More informationClass IX Chapter 1 Number Sustems Maths
Class IX Chapter 1 Number Sustems Maths Exercise 1.1 Question Is zero a rational number? Can you write it in the form 0? and q, where p and q are integers Yes. Zero is a rational number as it can be represented
More informationLecture 6.3: Polynomials and irreducibility
Lecture 6.3: Polynomials and irreducibility Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson)
More informationWORKSHEET ON NUMBERS, MATH 215 FALL. We start our study of numbers with the integers: N = {1, 2, 3,...}
WORKSHEET ON NUMBERS, MATH 215 FALL 18(WHYTE) We start our study of numbers with the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } and their subset of natural numbers: N = {1, 2, 3,...} For now we will not
More information= 5 2 and = 13 2 and = (1) = 10 2 and = 15 2 and = 25 2
BEGINNING ALGEBRAIC NUMBER THEORY Fermat s Last Theorem is one of the most famous problems in mathematics. Its origin can be traced back to the work of the Greek mathematician Diophantus (third century
More information3.4. ZEROS OF POLYNOMIAL FUNCTIONS
3.4. ZEROS OF POLYNOMIAL FUNCTIONS What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. Find rational zeros of polynomial functions. Find
More informationMATH10040: Numbers and Functions Homework 1: Solutions
MATH10040: Numbers and Functions Homework 1: Solutions 1. Prove that a Z and if 3 divides into a then 3 divides a. Solution: The statement to be proved is equivalent to the statement: For any a N, if 3
More informationTable of Contents. 2013, Pearson Education, Inc.
Table of Contents Chapter 1 What is Number Theory? 1 Chapter Pythagorean Triples 5 Chapter 3 Pythagorean Triples and the Unit Circle 11 Chapter 4 Sums of Higher Powers and Fermat s Last Theorem 16 Chapter
More informationFactor Rings and their decompositions in the Eisenstein integers Ring Z [ω]
Armenian Journal of Mathematics Volume 5, Number 1, 013, 58 68 Factor Rings and their decomositions in the Eisenstein integers Ring Z [ω] Manouchehr Misaghian Deartment of Mathematics, Prairie View A&M
More information2. Prime and Maximal Ideals
18 Andreas Gathmann 2. Prime and Maximal Ideals There are two special kinds of ideals that are of particular importance, both algebraically and geometrically: the so-called prime and maximal ideals. Let
More informationWinter Camp 2009 Number Theory Tips and Tricks
Winter Camp 2009 Number Theory Tips and Tricks David Arthur darthur@gmail.com 1 Introduction This handout is about some of the key techniques for solving number theory problems, especially Diophantine
More information1 Introduction. or equivalently f(z) =
Introduction In this unit on elliptic functions, we ll see how two very natural lines of questions interact. The first, as we have met several times in Berndt s book, involves elliptic integrals. In particular,
More informationLecture 7.5: Euclidean domains and algebraic integers
Lecture 7.5: Euclidean domains and algebraic integers Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley
More informationSolutions I.N. Herstein- Second Edition
Solutions I.N. Herstein- Second Edition Sadiah Zahoor Please email me if any corrections at sadiahzahoor@cantab.net. R is a ring in all problems. Problem 0.1. If a, b, c, d R, evaluate (a + b)(c + d).
More informationHow to prove it (or not) Gerry Leversha MA Conference, Royal Holloway April 2017
How to prove it (or not) Gerry Leversha MA Conference, Royal Holloway April 2017 My favourite maxim It is better to solve one problem in five different ways than to solve five problems using the same method
More informationProofs. Chapter 2 P P Q Q
Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,
More information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
More informationExercises on chapter 1
Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G
More informationBasic Logic and Proof Techniques
Chapter 3 Basic Logic and Proof Techniques Now that we have introduced a number of mathematical objects to study and have a few proof techniques at our disposal, we pause to look a little more closely
More informationSets and Motivation for Boolean algebra
SET THEORY Basic concepts Notations Subset Algebra of sets The power set Ordered pairs and Cartesian product Relations on sets Types of relations and their properties Relational matrix and the graph of
More informationFACTORING AFTER DEDEKIND
FACTORING AFTER DEDEKIND KEITH CONRAD Let K be a number field and p be a prime number. When we factor (p) = po K into prime ideals, say (p) = p e 1 1 peg g, we refer to the data of the e i s, the exponents
More informationRational Numbers CHAPTER. 1.1 Introduction
RATIONAL NUMBERS Rational Numbers CHAPTER. Introduction In Mathematics, we frequently come across simple equations to be solved. For example, the equation x + = () is solved when x =, because this value
More informationJUST THE MATHS UNIT NUMBER 1.3. ALGEBRA 3 (Indices and radicals (or surds)) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1 ALGEBRA (Indices and radicals (or surds)) by AJHobson 11 Indices 12 Radicals (or Surds) 1 Exercises 14 Answers to exercises UNIT 1 - ALGEBRA - INDICES AND RADICALS (or Surds)
More informationHowever another possibility is
19. Special Domains Let R be an integral domain. Recall that an element a 0, of R is said to be prime, if the corresponding principal ideal p is prime and a is not a unit. Definition 19.1. Let a and b
More informationChapter 3: Section 3.1: Factors & Multiples of Whole Numbers
Chapter 3: Section 3.1: Factors & Multiples of Whole Numbers Prime Factor: a prime number that is a factor of a number. The first 15 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,
More informationSolution Set 2. Problem 1. [a] + [b] = [a + b] = [b + a] = [b] + [a] ([a] + [b]) + [c] = [a + b] + [c] = [a + b + c] = [a] + [b + c] = [a] + ([b + c])
Solution Set Problem 1 (1) Z/nZ is the set of equivalence classes of Z mod n. Equivalence is determined by the following rule: [a] = [b] if and only if b a = k n for some k Z. The operations + and are
More informationArithmetic, Algebra, Number Theory
Arithmetic, Algebra, Number Theory Peter Simon 21 April 2004 Types of Numbers Natural Numbers The counting numbers: 1, 2, 3,... Prime Number A natural number with exactly two factors: itself and 1. Examples:
More informationa + b = b + a and a b = b a. (a + b) + c = a + (b + c) and (a b) c = a (b c). a (b + c) = a b + a c and (a + b) c = a c + b c.
Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called
More informationMath 131 notes. Jason Riedy. 6 October, Linear Diophantine equations : Likely delayed 6
Math 131 notes Jason Riedy 6 October, 2008 Contents 1 Modular arithmetic 2 2 Divisibility rules 3 3 Greatest common divisor 4 4 Least common multiple 4 5 Euclidean GCD algorithm 5 6 Linear Diophantine
More informationProperties of the Integers
Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called
More informationCourse 2316 Sample Paper 1
Course 2316 Sample Paper 1 Timothy Murphy April 19, 2015 Attempt 5 questions. All carry the same mark. 1. State and prove the Fundamental Theorem of Arithmetic (for N). Prove that there are an infinity
More information6 Cosets & Factor Groups
6 Cosets & Factor Groups The course becomes markedly more abstract at this point. Our primary goal is to break apart a group into subsets such that the set of subsets inherits a natural group structure.
More informationa + bi by sending α = a + bi to a 2 + b 2. To see properties (1) and (2), it helps to think of complex numbers in polar coordinates:
5. Types of domains It turns out that in number theory the fact that certain rings have unique factorisation has very strong arithmetic consequences. We first write down some definitions. Definition 5.1.
More informationLecture Notes. Advanced Discrete Structures COT S
Lecture Notes Advanced Discrete Structures COT 4115.001 S15 2015-01-13 Recap Divisibility Prime Number Theorem Euclid s Lemma Fundamental Theorem of Arithmetic Euclidean Algorithm Basic Notions - Section
More information2 Arithmetic. 2.1 Greatest common divisors. This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}.
2 Arithmetic This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}. (See [Houston, Chapters 27 & 28]) 2.1 Greatest common divisors Definition 2.16. If a, b are integers, we say
More informationIntroduction: Pythagorean Triplets
Introduction: Pythagorean Triplets On this first day I want to give you an idea of what sorts of things we talk about in number theory. In number theory we want to study the natural numbers, and in particular
More informationProposition The gaussian numbers form a field. The gaussian integers form a commutative ring.
Chapter 11 Gaussian Integers 11.1 Gaussian Numbers Definition 11.1. A gaussian number is a number of the form z = x + iy (x, y Q). If x, y Z we say that z is a gaussian integer. Proposition 11.1. The gaussian
More informationChapter 2. Polynomial and Rational Functions. 2.5 Zeros of Polynomial Functions
Chapter 2 Polynomial and Rational Functions 2.5 Zeros of Polynomial Functions 1 / 33 23 Chapter 2 Homework 2.5 p335 6, 8, 10, 12, 16, 20, 24, 28, 32, 34, 38, 42, 46, 50, 52 2 / 33 23 3 / 33 23 Objectives:
More informationTHE SPLITTING FIELD OF X 3 7 OVER Q
THE SPLITTING FIELD OF X 3 7 OVER Q KEITH CONRAD In this note, we calculate all the basic invariants of the number field K = Q( 3 7, ω), where ω = ( 1 + 3)/2 is a primitive cube root of unity. Here is
More informationRelative Densities of Ramified Primes 1 in Q( pq)
International Mathematical Forum, 3, 2008, no. 8, 375-384 Relative Densities of Ramified Primes 1 in Q( pq) Michele Elia Politecnico di Torino, Italy elia@polito.it Abstract The relative densities of rational
More information