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Binary Quadratic Forms, Genus Theory, and Primes of the Form P = X2 + Ny2

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Binary Quadratic Forms, Genus Theory, and Primes of the Form P = X2 + Ny2 Binary Quadratic Forms, Genus Theory, and Primes of the Form p = x2 + ny2 Josh Kaplan July 28, 2014 Contents 1 Introduction 1 2 Quadratic Reciprocity 2 3 Binary Quadratic Forms 5 4 Elementary Genus Theory 11 1 Introduction In this paper, we will develop the theory of binary quadratic forms and elemen- tary genus theory, which together give an interesting and surprisingly powerful elementary technique in algebraic number theory. This is all motivated by a problem in number theory that dates back at least to Fermat: for a given posi- tive integer n, characterizing all the primes which can be written p = x2 + ny2 for some integers x; y. Euler studied the problem extensively, and was able to solve it for n = 1; 2; 3; 4, giving the first rigorous proofs of the following four theorems of Fermat: p = x2 + y2 for some x; y 2 Z () p = 2 or p ≡ 1 mod 4; p = x2 + 2y2 for some x; y 2 Z () p = 2 or p ≡ 1; 3 mod 8; p = x2 + 3y2 for some x; y 2 Z () p = 3 or p ≡ 1 mod 3; p = x2 + 4y2 for some x; y 2 Z () p ≡ 1 mod 4: For n > 4, however, Euler was unsuccessful in giving a proof. He did manage conjectures, though, for many values of n, such as: p = x2 + 6y2 for some x; y 2 Z () p ≡ 1; 7 mod 24: With the techniques we will develop, these beautiful and, particularly for n = 6, otherwise difficult results follow from easy computations (as do similar results 1 for n = 5; 7; 9; 10; 12; 13, and over 50 other values of n). As we will see near the end of the paper, the techniques are not enough for us to characterize primes of the form p = x2 + ny2 for arbitrary n, but what they are able to achieve is remarkable in its own right. This paper will assume knowledge of basic field theory and group theory. 2 Quadratic Reciprocity In order to characterize primes of the p = x2 + ny2 by their congruences, we first need to prove the law of quadratic reciprocity. This law will allow us to easily determine when a number is a square in a finite field (remember that for prime p, Fp denotes the finite field Z=pZ). Just how crucial this ability is to our project will not become clear until the next section, but it is a beautiful and important theorem that has many uses outside of the problem we are focusing on and is worth proving for its own sake. To give our proof, we first need the following theorem. F∗ Theorem 2.1. Let p be an odd prime. Then an element x of p is a square if and only if x(p−1)=2 = 1. Proof. From Fermat's Little Theorem, we have: F∗ p−1 8a 2 p; a = 1: p−1 F∗ So the p − 1 roots of a − 1 are the distinct elements of p. This means that there are (p − 1)=2 distinct roots of a(p−1)=2 − 1 and of a(p−1)=2 + 1. If there F∗ 2 (p−1)=2 p−1 exists y in p such that y = x, then x = y = 1, so all squares in F∗ (p−1)=2 F∗ F∗ p are roots of a − 1. Since p is cyclic, there exists g in p such that F∗ 2 p−2 F∗ p = f1; g; g ; :::; g g, so there are at least (p − 1)=2 distinct squares in p. The theorem follows. F∗ Definition 2.2. Let p be an odd prime, and let x 2 p. The Legendre symbol x (p−1)=2 of x, denoted by p , is the integer x . Note that the Legendre symbol is frequently given the following alternative Z F∗ definition (in rather than p): 8 1; if [x] 2 F∗2; x 6≡ 0 mod p x < p = −1; if [x] 62 F∗2 p p : 0; if x ≡ 0 mod p: From Theorem 2.1, it is clear that for x 6≡ 0 mod p, this definition is equivalent to our own. A few properties of the Legendre symbol also follow immediately a b ab from our definition. Obviously, we have p p = p . Also, it is clear that −1 (p−1)=2 p = (−1) = 1 if and only if p ≡ 1 mod 4. 2 The following useful lemma is often referred to as Gauss's Lemma. It will allow us to prove the law of quadratic reciprocity, but it also will give us a 2 general formula for p . For the rest of this section, S will denote the subset p−1 F∗ F∗ f1; :::; 2 g of p, where p is an odd prime (note that p is the disjoint union F∗ of S and −S). Also, for any s in S, we will define es : p ! {±1g so that for F∗ all a in p, as = es(a)sa for some sa in S. F∗ a Q Lemma 2.3. Let p be an odd prime. Then for any a in p, p = s2S es(a): Proof. Clearly s 7! sa is a bijection from S onto itself, so we can use as = es(a)sa to get the following: ! ! ! ! (p−1)=2 Y Y Y Y Y a = es(a) sa = es(a) s : s2S s2S s2S s2S s2S Therefore a Y = a(p−1)=2 = e (a): p s s2S a v Note that this means p = (−1) where v is the number of times as falls in −S rather than S. And now, as promised, we can derive the following: 2 Proposition 2.4. p = 1 if and only if p ≡ 1; 7 mod 8 p−1 2 n(p) Proof. Since es(2) = 1 if and only if 2s ≤ 2 , we get p = (−1) , where p−1 p−1 n(p) is the number of integers s such that 4 < s ≤ 2 . Clearly p can be written in the form 4k ± 1, and we see n(p) = k. Then if p ≡ ±1 mod 8, k is even, and if p ≡ ±5 mod 8, k is odd, so the result is immediate. And we are also ready now to prove the major result of this section. For odd n, define 0; n ≡ 1 mod 4; E(n) = 1; n ≡ 3 mod 4: Theorem 2.5 (Law of Quadratic Reciprocity). Let l and p be two distinct odd l p E(p)E(l) primes. Then p = l (−1) . Proof. First, note the following trigonometric fact, which we will not prove here (although the proof is elementary): for any positive, odd integer m, (m−1)=2 sin mx Y 2πj = (−4)(m−1)=2 sin2 x − sin2 : sin x m j=1 3 F∗ F∗ Let S = f1; :::; (p − 1)=2g ⊆ p;T = f1; :::; (l − 1)=2g ⊆ l . Since ls = es(l)sl, of course, 2π 2π sin ls = e (l) sin s : p s p l So from Gauss's Lemma, we have: l Y Y 2πls 2πsl = e (l) = sin = sin : p s p p s2S s2S Using the fact that s 7! sl is a bijection, and the trigonometric fact we noted at the beginning of the proof, we get the following: l Y 2πls 2πs Y Y 2πs 2πt = sin = sin = (−4)(l−1)=2 sin2 − sin2 p p p p l s2S s2S t2T Y 2πs 2πt = (−4)(l−1)(p−1)=4 sin2 − sin2 : p l s2S;t2T If we permute l and p, the same argument gives us p Y 2πt 2πs = (−4)(l−1)(p−1)=4 sin2 − sin2 : l l p s2S;t2T Therefore l Y 2πt 2πs p = (−4)(l−1)(p−1)=4 − sin2 − sin2 = (−1)(p−1)(l−1)=4 p l p l s2S;t2T p = (−1)E(p)E(l) : l The importance of this result for characterizing primes of the form p = x2 + ny2 will soon become clear. For now, just note that the law of quadratic reciprocity, when combined with the multiplicative property of Legendre sym- bols, gives us the remarkable ability to determine whether any given element of F∗ p is a square without having to find its square root. We will give a few exam- ples here of Legendre symbol calculations before returning to our main project in the next section. Example 2.6. 137 307 33 3 11 137 137 2 5 = = = = = 307 137 137 137 137 3 11 3 11 2 11 = 3 5 2 1 = 3 5 = −1 ∗ 1 = −1: 4 89 509 64 2 8 8 Example 2.7. 509 = 89 = 89 = 89 = 1, since (±1) = 1. 3 Binary Quadratic Forms At the end of this section, we will be able to use the theory of reduced forms to characterize primes of the form p = x2 + ny2 for n = 1; 2; 3; 4; 7. This theory is elementary but quite powerful, and to develop it, we first must define some concepts which will serve as the foundation for the rest of our discussion. Definition 3.1. A binary quadratic form (hereafter just quadratic form) is a function in two variables f(x; y) = ax2 + bxy + cy2. Our discussion will be limited to integral quadratic forms (i.e. a; b; c 2 Z). We say a quadratic form f(x; y) is primitive if a; b and c are relatively prime. Definition 3.2. An integer m is represented by quadratic form f if there exist x; y in Z such that f(x; y) = m.
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