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Class Field Theory and the Theory of N-Fermat Primes

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Class Field Theory and the Theory of N-Fermat Primes CLASS FIELD THEORY AND THE THEORY OF N-FERMAT PRIMES BY ANDREW KOBIN A Thesis Submitted to the Graduate Faculty of WAKE FOREST UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES in Partial Fulfillment of the Requirements for the Degree of MASTER OF ARTS Mathematics May, 2015 Winston-Salem, North Carolina Approved By: Frank Moore, Ph.D., Advisor Hugh Howards, Ph.D., Chair Jeremy Rouse, Ph.D. Acknowledgments This was probably the hardest page of this thesis to write, as no number of words are sufficient to praise those who have helped and supported me along the way to my Master's degree. First, I would like to thank the members of my committee. Thank you to Dr. Jeremy Rouse for his seemingly infinite wisdom and even greater generosity in sharing his knowledge with me. We had numerous discussions on the finer details of algebraic number theory that helped shape the direction of my research. He is also a fast runner. Thank you to Dr. Hugh Howards for his mentorship and advice going all the way back to 2010 when I was mere freshman at Wake Forest. My identity as a mathematician is in large part due to the dedication of Dr. Howards as an educator. Lastly, thank you to my adviser, Dr. Frank Moore, for his selfless devotion to this project over nearly two years' time. I would not be where I am today without his mathematical knowledge, worldly advice and sincere friendship. The Department of Mathematics at Wake Forest has been like a second family to me for some years now. Thank you to everyone here that has helped me to survive and thrive at Wake Forest. To my office mates, Mackenzie, Elliott, Elena and Amelie: thank you for your support and for putting up with my loud music! Finally, I would like to thank my family for their love and for providing me with opportunities in life that have allowed me to succeed. They are my biggest supporters and I love them dearly. ii Table of Contents Acknowledgments . ii Abstract . v Chapter 1 Algebraic Number Fields . 1 1.1 Rings of Algebraic Integers . .1 1.2 Dedekind Domains . .5 1.3 Ramification of Primes . 12 1.4 The Decomposition and Inertia Groups . 18 1.5 Norms of Ideals . 22 1.6 Discriminant and Different . 26 1.7 The Class Group . 34 1.8 The Hilbert Class Field . 45 1.9 Orders . 58 1.10 Units in a Number Field . 70 Chapter 2 Class Field Theory . 78 2.1 Valuations and Completions . 78 2.2 Frobenius Automorphisms and the Artin Map . 90 2.3 Ray Class Groups . 96 2.4 L-series and Dirichlet Density . 105 2.5 The Frobenius Density Theorem . 118 2.6 The Second Fundamental Inequality . 126 2.7 The Artin Reciprocity Theorem . 134 2.8 The Conductor Theorem . 143 2.9 The Existence and Classification Theorems . 145 2.10 The Cebotarevˇ Density Theorem . 150 2.11 Ring Class Fields . 160 Chapter 3 Quadratic Forms and n-Fermat Primes . 168 3.1 The Theory of Binary Quadratic Forms . 168 3.2 The Form Class Group . 175 3.3 n-Fermat Primes . 183 iii Bibliography . 190 Appendix A Appendix. 193 A.1 The Four Squares Theorem . 193 A.2 The Snake Lemma . 195 A.3 Cyclic Group Cohomology . 198 A.4 Helpful Magma Functions . 201 Curriculum Vitae . 209 iv Abstract Andrew J. Kobin Most problems in number theory are exceedingly simple to state, yet many con- tinue to elude mathematicians even centuries after they were originally posed. Such a question, \Given a positive integer n, when can a prime number be written in the form x2 + ny2?", was solved by Cox [7], and although the statement is elementary, the solution requires the depth and power of class field theory to understand. In our approach to this question, we will explore a variety of topics, including: algebraic number fields; types of class groups and class fields; two density theorems; the main theorems in class field theory; and the theory of quadratic forms. Our discussion will culminate in Theorem 2.11.3, a full characterization of primes of the form x2 + ny2. However, the intrigue doesn't end there. In Chapter 3, we pose the related ques- tion: \If p is a prime of the form x2 +ny2, when is y2 +nx2 also prime?" This question turns out to be much harder to approach, but we will investigate the symmetric n- Fermat prime question thoroughly. In certain sections (1.8, 2.10 and 3.3) we use the Magma Computational Alge- bra System to handle large or complicated computations. Many of the basic com- mands can be found in the Magma handbook, available at http://magma.maths. usyd.edu.au/magma/handbook/ through the University of Sydney's Computational Algebra Group. v Chapter 1: Algebraic Number Fields In the first chapter we provide a detailed description of the main topics in algebraic number theory: algebraic number fields, rings of integers, the behavior of prime ideals in extensions, norms of ideals, the discriminant and different, the class group, the Hilbert class field, orders and Dirichlet's unit theorem. 1.1 Rings of Algebraic Integers Let Q be an algebraic closure of Q. Then Q is an infinite dimensional Q-vector space and every polynomial f 2 Q[x] splits in Q[x]. An example of such an algebraic closure is Q = fu 2 C j f(u) = 0 for some f 2 Q[x]g. Then Q ⊂ Q ⊂ C. Note that any two choices of Q are isomorphic. One of the most important elements of a number field we will be working with is: Definition. An element α 2 Q is an algebraic integer if it is a root of some monic polynomial with coefficients in Z. p 2 1 Example 1.1.1. 2 is an algebraic integer since it is a root of x − 2. However, 2 ; π 1 and e are not algebraic integers. We will see in a moment why 2 is not algebraic, but the proof for π and e is famously difficult. Note that the set of algebraic integers in Q is precisely the integers Z. In a moment we will generalize this set to fields other than Q. Definition. The minimal polynomial of α 2 Q is the monic polynomial f 2 Q[x] of minimal degree such that f(α) = 0. The minimal polynomial of α is unique, as the following lemma shows. 1 Lemma 1.1.2. Suppose α 2 Q. Then the minimal polynomial f of α divides any other polynomial h such that h(α) = 0. Proof. Suppose h(α) = 0. Then by the division algorithm, h = fq + r with deg r < deg f. Note that r(α) = h(α) − f(α)q(α) = 0 so α is a root of r. But since deg f is minimal among all polynomials of which α is a root, r must be 0. This shows that f divides h. Lemma 1.1.3. If α 2 Q is an algebraic integer then the minimal polynomial has coefficients in Z. Proof. Let f 2 Q[x] be the minimal polynomial of α. Since α is an algebraic integer, there is some g 2 Z[x] such that g(α) = 0. By Lemma 1.1.2, g = fh for some monic h 2 Q[x]. Suppose f 62 Z[x]. Then there is some prime p dividing the denominator of at least one of the coefficients of f; let pi be the largest power of p that divides a denominator. Likewise let pj be the largest power of p that divides the denominator of a coefficient of h. Then pi+jg = (pif)(pjh) and reducing mod p gives 0 on the left, but two nonzero polynomials in Fp[x] on the right, a contradiction. Hence f 2 Z[x]. An important characterization of algebraic integers is provided in the following proposition. Proposition 1.1.4. α 2 Q is an algebraic integer if and only if ( n ) X i Z[α] = ciα : ci 2 Z; n ≥ 0 i=0 is a finitely generated Z-module. Proof. ( =) ) Suppose α is integral with minimal polynomial f 2 Z[x], where deg f = k. Then Z[α] is generated by 1; α; : : : ; αk−1. 2 ( ) = ) Suppose α 2 Q and Z[α] is generated by f1(α); : : : ; fn(α). Let d ≥ M where M = maxfdeg fi j 1 ≤ i ≤ ng. Then n d X α = aifi(α) i=1 n d X for some choice of ai 2 Z. Hence α is a root of x − aifi(x) so it is integral. i=1 1 1 Example 1.1.5. α = 2 is not an algebraic integer since Z 2 is not finitely generated as a Z-module. Definition. For a given algebraic closure Q of Q, we will denote the set of all algebraic integers in Q by Z. This set inherits the binary operations + and · from Q, and an important property is that Z is closed under these operations: Proposition 1.1.6. The set Z of all algebraic integers is a ring. Proof. Note that 0 is a root of the zero polynomial, so 0 2 Z. Then it suffices to prove closure under addition and multiplication. Suppose α; β 2 Z and let m and n be the degrees of their respective minimal polynomial. Then 1; α; : : : ; αm−1 span Z[α] and 1; β; : : : ; βn−1 likewise span Z[β]. So the elements αiβj for 1 ≤ i ≤ m; 1 ≤ j ≤ n span Z[α; β], so this Z-module is finitely generated.
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