Number Theory

Download e-book for iPad: Advances in number theory: Proc. 3rd conf. of Canadian by Fernando Q. Gouvea, Noriko Yui

By Fernando Q. Gouvea, Noriko Yui

This ebook includes the lawsuits of the 3rd convention of the Canadian quantity idea organization. The 38 technical papers awarded during this quantity speak about correct and well timed concerns within the fields of analytic quantity concept, arithmetical algebraic geometry, and diophantine approximation. The publication contains a number of papers honoring Paulo Ribenboim, to whom this convention used to be devoted.

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Additional info for Advances in number theory: Proc. 3rd conf. of Canadian Number Theory Association, 1991

Example text

D) If J = Z[X] prove that G(X) is irreducible in Q[X], and conversely that if G(X) is irreducible then I = G(X)Z[X] is a prime ideal. (e) Finally, we assume from now on that J = Z[X], or equivalently, that G(X) ∈ / I, hence that J is a prime ideal. Prove that J ⊂ I, hence that I = J, and deduce that G(X) = 1. (f) Prove that the integer n defined above is a prime number p, and in particular that n = 1. (g) Prove that there exists a polynomial H(X) ∈ Z[X] such that I = pZ[X] + H(X)Z[X], and that the reduction H(X) in Fp [X] is either 0 or is irreducible in Fp [X].

D) If J = Z[X] prove that G(X) is irreducible in Q[X], and conversely that if G(X) is irreducible then I = G(X)Z[X] is a prime ideal. (e) Finally, we assume from now on that J = Z[X], or equivalently, that G(X) ∈ / I, hence that J is a prime ideal. Prove that J ⊂ I, hence that I = J, and deduce that G(X) = 1. (f) Prove that the integer n defined above is a prime number p, and in particular that n = 1. (g) Prove that there exists a polynomial H(X) ∈ Z[X] such that I = pZ[X] + H(X)Z[X], and that the reduction H(X) in Fp [X] is either 0 or is irreducible in Fp [X].

Since g is a primitive root, we can write a ≡ g x (mod pv ), so that v −1 ≡ g p x (mod pv ), and since g has order pv−1 (p − 1), it is clear that x a v −1 is unique modulo p−1. Since ap −1 ≡ 1 (mod p) by Fermat’s little theorem, we must simply show that for any b ≡ 1 (mod p) there exists y such that b ≡ (1 + p)y (mod pv ). 1 Finitely Generated Abelian Groups 25 (Z/pv Z)∗ of elements congruent to 1 modulo p. 22, it follows that it is bijective, showing the existence of y and its uniqueness modulo pv−1 .

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