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Arithmetic of Cyclotomic Fields

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Arithmetic of Cyclotomic Fields Arithmetic of cyclotomic fields Tudor Ciurca September 26, 2018 ∗ Contents 1 Dedekind domains and their ideals 4 1.1 Rings of integers are Dedekind domains . 4 1.2 Unique prime factorization (UPF) of ideals in Dedekind domains . 6 1.3 Ideal factorization . 9 1.4 Decomposition of primes in field extensions. 14 1.5 Orders of number fields in general . 18 1.6 More on prime decomposition . 26 1.7 More on discriminants . 31 1.8 The different ideal . 35 2 Examples of prime decomposition in number fields 41 2.1 Prime decomposition in quadratic fields . 41 2.2 Prime decomposition in pure cubic fields . 43 2.3 Prime decomposition in cyclotomic fields . 47 2.4 Cubic fields in general . 50 2.5 Quadratic reciprocity via prime decomposition . 54 3 Ring of adeles of a number field 56 3.1 Definitions of adeles and ideles . 56 3.2 Compactness of the reduced idele class group . 64 3.3 Applications to finiteness of ideal class group and Dirichlet's unit theorem . 68 ∗Department of Mathematics, Imperial College London, London, SW7 6AZ, United Kingdom E-mail address: [email protected] 1 4 L-series and zeta functions 71 4.1 Definitions and first properties . 71 4.2 Dirichlet's theorem on arithmetic progressions . 74 4.3 The analytic class number formula . 78 4.4 Applications and examples of the analytic class number formula . 85 4.5 Dirichlet characters and associated number fields . 88 5 Arithmetic of cyclotomic fields and Fermat's last theorem 91 5.1 Arithmetic of cyclotomic fields . 91 5.2 Case 1 of Fermat's last theorem . 95 5.3 Case 2 of Fermat's last theorem . 96 5.4 Cases p = 3 and p =4................................... 98 5.5 The relative class number formula for prime cyclotomic fields . 99 6 More arithmetic of cyclotomic fields 108 6.1 Construction of p-adic L-functions . 108 6.2 Gauss sums and the Stickelberger relation . 120 6.3 Herbrand's theorem . 127 6.4 Kummer's criterion for the regularity of primes . 129 7 Acknowledgements 133 2 Abstract For 300 years since its conception, Fermat's last theorem went unsolved. At first, only special cases were demonstrated. Fermat himself proved the exponent 4 case. Euler proved the exponent 3 case. Dirichlet proved the exponent 5 case. Lam´e'sproof for the exponent 7 case was quite complicated. Most of these early proofs went by descent. It was Lam´ewho first noticed that cyclotomic fields are the right environment to work with in regards to Fermat's last theorem. He showed that Fermat's last theorem for exponent p would follow from unique prime factorization in the ring Z[ζp], but he incorrectly assumed that unique prime factorization holds in all such rings. Here is where Kummer comes into the picture. He arguably had the most impact on Fermat's last theorem before Wiles. Kummer introduced the concept of ideal numbers, a precursor to ideals, in order to fix unique prime factorization when it fails. He proved that ideal numbers have unique prime factorization, and fixed Lame's proof, even though it was limited to prime exponents satisfying a certain condition. These primes were called regular. Statistically, around 61% of primes are regular, although the infinitude of regular primes is still an unsolved problem. This document is the result of a UROP project undertaken in the summer of 2018 at Imperial College London. The preliminaries assumed in this document include definitions from an undergraduate course in commutative algebra and results from Galois theory. In later chapters we make use of class field theory and Kummer theory, and refer the reader to some sources where this theory is developed. Section 1 provides a comprehensive introduction to some topics in algebraic number theory at the level of an advanced undergraduate course. Section 2 uses the theory developed in Section 1 to study specific examples and families of number fields. Section 3 gives a crash course on the theory of adeles, assuming some knowledge of local fields. This will be used to prove Dirichlet's unit theorem and the finiteness of the ideal class group. Section 4 introduces Dirichlet L-series and proves Dirichlet's theorem on primes in arith- metic progression. We also introduce Dedekind zeta functions and derive the analytic class number formula. We then study the relationship between Dirichlet characters of abelian Ga- lois groups and their associated intermediate number fields. Section 5 is where we actually begin studying the arithmetic of cyclotomic fields. We apply our results to prove Fermat's last theorem for regular primes, following Kummer. We then build upon results in Section 4 and derive the relative class number formula for prime cyclotomic fields. In Section 6, our main aim is to prove Kummer's criterion for the regularity of a prime. We give both an analytic and algebraic derivation for the preliminary criterion of whether p divides the relative class number. The analytic method uses p-adic L-functions whilst the algebraic method uses the Stickelberger relation to prove Herbrand's theorem. Along the way we study Gauss sums which will hep us prove Stickelberger's relation. Finally we use Kummer theory to relate the relative class number to the class number of a prime cyclotomic field. 3 1 Dedekind domains and their ideals Number theory began as the study of the integers. They act as a scaffolding for the field of rational numbers. During the 19th century mathematicians realized that number fields, that is, field extensions of Q of finite degree, have similar properties to Q and are the right context to study solutions of polynomial equations with integer coefficients in one variable. The "number theory" of a number field can be developed in a similar way to that of Q. There is an analogue to the integers for every number field, called its ring of integers, which acts as the scaffolding of the number field and contains many of its number-theoretic properties. In this section we will study number fields and their rings of integers. The important properties satisfied by these rings of integers are summarized in the abstract object known as a Dedekind domain, which is defined below. In this document every ring is commutative and has a multiplicative identity. The material in this section is based on a variety of sources. The main source is a course in algebraic number theory that the author has undertaken at the summer school PROMYS Europe 2017. This is also supplemented by [Cox13] and several expository papers by Keith Conrad and William Stein. At the beginning of every subsection, the specific sources used will be mentioned. If not, then the material of the subsection originates from the algebraic number theory course that the author took. Definition 1.0.1. A Dedekind domain is an integral domain R which is also • Noetherian • of Krull dimension 1 • integrally closed in Frac(R), the fraction field of R 1.1 Rings of integers are Dedekind domains Why are we interested in Dedekind domains? It is because the ring of integers OK of a number field K is a Dedekind domain. Recall that the ring of integers is the integral closure of Z in K, and these are objects of central importance in algebraic number theory. The first two conditions above can be thought of as some sort of restriction on the size of the ring. Note that one does not imply the other; there are examples of Krull dimension 1 rings which are not Noetherian. The next lemma proves that the rank of the ring of integers as a Z-module equals the vector space dimension of the number field over Q. This will be the crucial step in proving that rings of integers satisfy the first two conditions. Lemma 1.1.1. Let K be a number field. Then OK is a free Z − module of rank [K : Q]. Proof. Let K = Q(a1 : : : an) so that fa1 : : : ang is a Q-basis for K. We will show that the ai can Pm j be chosen to be elements of OK . Assume ai is not such an element and let fi = j=0 bi;jx be its 4 m−1 minimal polynomial with bi;j integers and bi;m non-zero. Then we can multiply fi by bi;m to get m m−1 X m−j−1 j bi;m fi = bi;jbi;m (bi;mai) = 0 j=0 m−1 −1 As a result the polynomial bi;m fi(bi;mx) is monic and has a root bi;mai which is an element of OK . Hence we can replace ai with bi;mai since bi;m is a non-zero integer. Now Z[a1 : : : an] ⊂ OK is a free Z-module of rank n, because the a1 : : : an being linearly inde- pendent over Q implies that they are linearly independent over Z. To show that OK is also a free Z-module of rank n, we will consider the embedding n : OK ! Z : g 7! (T r(g · a1) : : : T r(g · an)) where T r(·) is the absolute trace in K. This is clearly a Z-module homomorphism. If each T r(g ·ai) is zero then T r(g · h) = 0 for any h 2 K since the trace is Q-linear and fa1 : : : ang is Q-basis for K. N(g) In particular that means T r(N(g)) = 0 by selecting h = g , where N(·) is the absolute norm in K. This implies that g = 0 because N(g) is an integer and the trace of integers satisfies T r(N(g)) = n · N(g) This shows that is injective, and so it is indeed an embedding.
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