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Notes on the Theory of Algebraic Numbers

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Notes on the Theory of Algebraic Numbers Notes on the Theory of Algebraic Numbers Steve Wright arXiv:1507.07520v1 [math.NT] 27 Jul 2015 Contents Chapter 1. Motivation for Algebraic Number Theory: Fermat’s Last Theorem 4 Chapter 2. Complex Number Fields 8 Chapter 3. Extensions of Complex Number Fields 14 Chapter 4. The Primitive Element Theorem 18 Chapter 5. Trace, Norm, and Discriminant 21 Chapter 6. Algebraic Integers and Number Rings 30 Chapter 7. Integral Bases 37 Chapter 8. The Problem of Unique Factorization in a Number Ring 44 Chapter 9. Ideals in a Number Ring 48 Chapter 10. Some Structure Theory for Ideals in a Number Ring 57 Chapter 11. An Abstract Characterization of Ideal Theory in a Number Ring 62 Chapter 12. Ideal-Class Group and the Class Number 65 Chapter 13. Ramification and Degree 71 Chapter 14. Ramification in Cyclotomic Number Fields 81 Chapter 15. Ramification in Quadratic Number Fields 86 Chapter 16. Computing the Ideal-Class Group in Quadratic Fields 90 Chapter 17. Structure of the Group of Units in a Number Ring 100 Chapter 18. The Regulator of a Number Field and the Distribution of Ideals 119 Bibliography 124 Index 125 3 CHAPTER 1 Motivation for Algebraic Number Theory: Fermat’s Last Theorem Fermat’s Last Theorem (FLT). If n 3 is an integer then there are no positive ≥ integers x, y, z such that xn + yn = zn. This was first stated by Pierre de Fermat around 1637: in the margin of his copy of Bachet’s edition of the complete works of Diophantus, Fermat wrote (in Latin): “It is impossible to separate a cube into two cubes or a bi-quadrate into two bi-quadrates, or in general any power higher than the second into powers of like degree; I have discovered a truly remarkable proof which this margin is too small to contain.” FLT was proved (finally!) by Andrew Wiles in 1995. One can show FLT if and only if one can prove FLT for primes. If p is an odd prime then there are no nonzero integers x, y, z, such that xp + yp + zp =0. Modern algebraic number theory essentially began in an attack on FLT by the great German number theorist Ernst Eduard Kummer in 1840. In order to explain Kummer’s strategy, we need to recall the notion of a unique factorization domain (UFD). Definitions. Let D be an integral domain, i.e., a commutative ring with identity 1 that contains no zero divisors, i.e., no elements x, y such that x =0 = y and xy = 0. 6 6 (i) if a, b D and if there exists c D such that b = ac, then a divides b, or a is a factor ∈ ∈ of b, denoted a b. | (ii) u D is a unit in D if u has a mulitiplicative inverse in D. ∈ If U(D) denotes the set of all units in D then 1, 1 U(D) and U(D) is an abelian {− } ⊆ group under multiplication in D; U(D) is the group of units in D. (iii) a, b D are associates if there exists u U(D) such that a = bu. ∈ ∈ (iv) 0 = p D is prime (or irreducible) if p U(D) and p = ab for a, b D implies that 6 ∈ 6∈ ∈ either a or b is a unit. 4 1. MOTIVATION FOR ALGEBRAIC NUMBER THEORY: FERMAT’S LAST THEOREM 5 1 If a D, u U(D) then a = (au− )u, i.e., every element of D has a factorization of ∈ ∈ the form (unit) (element of D). Such factorizations are hence said to be trivial. Primes in × D are precisely the elements of D with only trivial factorizations. If Z denotes the ordinary ring of integers then U(Z)= 1, 1 , {− } and the primes in Z, according to (iv) in the above definitions, are precisely the prime numbers, together with their negatives. Hence a prime in an integral domain is the analog of a prime number in Z. N.B. In order to avoid ambiguity that may arise from the terminology with regard to primes that we have introduced, we will refer to a positive prime number in Z as a rational prime. Definition. An integral domain D is a unique factorization domain (UFD) if (i) every element in D 0 U(D) can be factored into a product of primes in D, and \ { } ∪ (ii) If p p and q q are factorizations into primes of the same element of D then 1 ··· r 1 ··· s r = s and the qj’s can be reindexed so that pi and qi are associates for i =1,...,r. Kummer’s strategy for proving Fermat’s Last Theorem Assume that p is an odd rational prime and that there exits nonzero integers x, y, z such that ( ) xp + yp = zp. ∗ We want to derive a contradiction from this assumption. In order to do this Kummer split the situation into the following two cases: Case I: p divides none of x, y, z. Case II: p divides at least one of x, y, z. We will discuss what Kummer did only for Case I. Notation. In the sequel, if S is a set and n is a positive integer then Sn will denote the Cartesian product of S with itself taken n times, i.e., the set of all n-tuples (s1,...,sn), where s S for all i. i ∈ It’s easy to derive a contradiction for p = 3. If neither x, y, nor z is divisible by 3, then x3, y3 and z3 are each 1 mod 9, and so x3 + y3 2, 0, or 2 mod 9, whence x3 + y3 z3 ≡ ± ≡ − 6≡ mod 9, contrary to ( ) with p = 3. ∗ Hence suppose that p> 3. Let 2π 2π ω = ω = e2πi/p = cos + i sin , p p p 1. MOTIVATION FOR ALGEBRAIC NUMBER THEORY: FERMAT’S LAST THEOREM 6 a p-th root of unity. It can be shown (see Proposition 25, infra) that the set Z[ω] of complex numbers defined by p 2 − i p 1 Z[ω]= aiω :(a0,...,ap 2) Z − − ∈ ( i=0 ) X is a subring of the set of complex numbers C, i.e., Z[ω] is closed under addition, subtraction, and multiplication of complex numbers, and is also clearly an integral domain. Now, suppose that Z[ω] is a UFD. Kummer then proved that ( ) there exists a unit u Z[ω] and α Z[ω] such that x + yω = uαp. ∗∗ ∈ ∈ He then used ( ), ( ), and the assumption that p does not divide x or y (Case I) to show ∗ ∗∗ that x y mod p. ≡ Applying the same argument using xp +( z)p =( y)p, he also got that − − x z mod p. ≡ − But then 2xp xp + yp = zp xp mod p, ≡ ≡ − and so 3xp 0 mod p, ≡ i.e., p (3xp), hence p x or p 3. Because p > 3, it follows that p x, and this contradicts the | | | | hypothesis of Case I. Thus Kummer had shown that if Z[ω] is a UFD then Case I cannot be true, i.e., if ( ) is true and Z[ω] is a UFD then p must divide at least one of x, y, or z. Kummer ∗ was thus led to ask is Z[ωp] a UFD, for all rational primes p> 3? The answer, unfortunately, is no: Kummer was able to prove that Z[ω23] is in fact not a UFD. So the next question must be for what p is Z[ωp] a UFD? Answer: all p 19 and no others! This is very difficult to prove, and was not done until ≤ 1971. For some discussion of the ideas that Kummer used to derive a contradiction in Case I when Z[ωp] is not a unique factorization domain, see Marcus [9], Chapter 1. 1. MOTIVATION FOR ALGEBRAIC NUMBER THEORY: FERMAT’S LAST THEOREM 7 Let Q denote the set of all rational numbers, and let Q[ωp] denote the set of complex numbers defined by p 2 − i p 1 Q[ωp]= aiωp :(a0,...,ap 2) Q − . − ∈ ( i=0 ) X Clearly Z[ωp] Q[ωp], and we will eventually prove that Q[ωp] is a subfield of C, i.e., Q[ωp] ⊆ is closed under addition, subtraction, multiplication, and division of complex numbers. It turns out that arithmetic properties of Z[ωp] such as unique factorization and the existence of units with useful algebraic properties are closely tied to algebraic properties of Q[ωp]; indeed, much of Kummer’s own work in number theory turns on a deep study of this connection. We are hence arrived at the fundamental questions of algebraic number theory: (a) What are the subfields F of C which have a distinguished subring R such that (i) the arithmetic, i.e., ring-theoretic, structure of R can be used to solve interesting and important problems in number theory and such that (ii) the arithmetic structure of R can be effectively studied by means of the field-theoretic structure of F ? (b) Given a class of fields F and subrings R of F which answer question (a), what is the mathematical technology which can be used to get the ring-theoretic structure of R from the field-theoretic structure of F ? We will spend our time in these notes getting some good answers to these very important questions. CHAPTER 2 Complex Number Fields Definition. A complex number field is a nonempty set F of complex numbers such that F = 0 and F is closed under addition, subtraction, multiplication, and division of complex 6 { } numbers, i.e., F is a nonzero subfield of C.
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