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NT2009qrproof

# NT2009qrproof - QUADRATIC RECIPROCITY II THE PROOFS PETE L...

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QUADRATIC RECIPROCITY II: THE PROOFS PETE L. CLARK We shall prove the Quadratic Reciprocity Law and its “second supplement.” 1. Preliminaries on congruences in cyclotomic rings For a positive integer n , let ζ n = e 2 πi n be a primitive n th root of unity, and let R n = Z [ ζ n ] = { a 0 + a 1 ζ n + . . . + a n - 1 ζ n - 1 n | a i Z } . Recall that an algebraic integer is a complex number α which satisfies a monic polynomial relation with Z -coefficients: there exist n and a 0 , . . . , a n - 1 such that α n + a n - 1 α n - 1 + . . . + a 1 α + a 0 . We need the following purely algebraic fact: Proposition 1. The algebraic integers form a subring of the complex numbers. This amounts to showing that if α and β are algebraic integers, then α + β and α · β are algebraic integers (which is plausible but not so trivial to prove). We have relegated the proof to [Integral Elements and Extensions]. Let p be a prime number; for x, y R n , we will write x y (mod p ) to mean that there exists a z R n such that x - y = pz . Otherwise put, this is congruence modulo the principal ideal pR n of R n . Since Z R n , if x and y are ordinary integers, the notation x y (mod p ) is ambiguous: interpreting it as a usual congruence in the integers, it means that there exists an integer n such that x - y = pn ; and interpreting it as a congruence in R n , it means that x - y = pz for some z R n . The key technical point is that these two notions of congruence are in fact the same: Lemma 2. If x, y Z and z R n are such that x - y = pz , then z Z . Proof: Just dividing by p , we find that the complex number z = x - y p is visibly an element of Q . Now we need the fact, proved in Handout 2, that the only algebraic integers which are rational numbers are the usual integers. (This is the reason why we needed to assert that every element of R n was an algebraic integer.) To prove the second supplement we will take n = 8. To prove the QR law we will take n = p an odd prime. These choices will be constant throughout each of the proofs so we will abbreviate ζ = ζ 8 (resp. ζ p ) and R = R 8 (resp. R p ). 2. Proof of the Second Supplement Put ζ = ζ 8 , a primitive eighth root of unity and R = R 8 = Z [ ζ 8 ]. We have: 0 = ζ 8 - 1 = ( ζ 4 + 1)( ζ 4 - 1) . 1

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2 PETE L. CLARK Since ζ 4 = 1 (primitivity), we must have ζ 4 + 1 = 0. Multiplying by ζ - 2 we get ζ 2 + ζ - 2 = 0 .
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NT2009qrproof - QUADRATIC RECIPROCITY II THE PROOFS PETE L...

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