Number Theory: An Introduction to Mathematics
126 II Divisibility We have developed the arithmetic of quaternions only as far as is needed to prove the four-squares theorem. ...
Additional References 127 [23] S. Lang,Algebra, corrected reprint of 3rd ed., Addison-Wesley, Reading, Mass., 1994. [24] D.H. Le ...
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III More on Divisibility In this chapter the theory of divisibility is developed further. The various sections of the chapter ar ...
130 III More on Divisibility is a permutation ofX.WedefinetheJacobi symbol(a/n)to be sgn(πa),i.e. (a/n)=1or− 1 according as the ...
1 The Law of Quadratic Reciprocity 131 and henceα:=vμ−^1 has sign sgn(α)=(− 1 )(m+n)(m−^1 )(n−^1 )/^2 (m/n)m(n/m)n. Butαis the p ...
132 III More on Divisibility where the minus sign holds if and only ifnandε 1 n 1 are both congruent to 3 mod 4. The process can ...
1 The Law of Quadratic Reciprocity 133 Letgbe a primitive root modp. Then the integers 1,g,...,gp−^2 modpare just a rearrangemen ...
134 III More on Divisibility Proof Suppose first thata=−1. Since(− 1 /p)=(− 1 )(p−^1 )/^2 ,wewishtoshow that there are infinitel ...
1 The Law of Quadratic Reciprocity 135 A second proof of the law of quadratic reciprocity will now be given. Letpbe an odd prime ...
136 III More on Divisibility or, puttingv=uw, τ^2 = ∑ w≡0modp (w/p) ∑ u≡0modp ζu(^1 +w). Since the coefficients oftp−^1 andtp− ...
1 The Law of Quadratic Reciprocity 137 G(m,n)= n∑− 1 v= 0 e^2 πiv (^2) m/n . Instead of summing from 0 ton−1 we can just as well ...
138 III More on Divisibility by Dirichlet’s convergence criterion in the theory of Fourier series, {F(+ 0 )+F(− 0 )}/ 2 = lim N→ ...
1 The Law of Quadratic Reciprocity 139 whereCis theFresnel integral C= ∫∞ −∞ e^2 πit 2 dt. (This is an important example of an i ...
140 III More on Divisibility Then ρ(n,m)=ρ(r,m)=(r/m)=(n/m). Sinceρ(m,n)=(m/n), this yields a contradiction. Thus,if n is an od ...
2 Quadratic Fields 141 N(α)=αα′=r^2 −ds^2. EvidentlyN(α)=N(α′),andN(α)=0 if and only ifα=0. From the relation (αβ)′=α′β′we obtai ...
142 III More on Divisibility We have already seen in§6 of Chapter II that the ringGof Gaussian integers is a Euclidean domain, w ...
2 Quadratic Fields 143 it follows that x+y= 9 a^3 , x^2 −xy+y^2 = 3 b^3 , wherea,b∈Zand 3b. We now shift operations to the Eucl ...
144 III More on Divisibility The proof of Proposition 12 illustrates howproblems involvingordinary integers may be better unders ...
2 Quadratic Fields 145 a,b∈N. Consequently there is a least unitε 0 >1. Then, for any unitε>1, there is a positive integer ...
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