An analogue of Bridges and Mena%26apos;s theorem for local fields and a local-global principle

作者：Krakovski Roi^*

来源：Linear Algebra and Its Applications, 2013, 438(1): 342-347.

DOI：10.1016/j.laa.2012.07.043

• p-adic numbers
• Character sums
• Abelian groups
• Minimal polynomials
• Local-global principle

Let G be an abelian group of finite order n, K a field and R subset of K a ring. Let D = Sigma(g is an element of G) a(g)g is an element of R[G] such that chi (D) is an element of R for every character chi : G -%26gt; K(xi(n)) (where chi (D) = Sigma(g is an element of G) a(g)chi (g) and xi(n) is a primitive nth root of unity). What does D look like? The case where K = Q and R = Z was settled by Bridges and Mena. Here we obtain a complete characterization for the case where K is a finite extension of the field Q(p) and R is its valuation ring under the condition that p does not divide n. %26lt;br%26gt;As an application we obtain the following local-global principle for Z/q(1)q(2)Z (where q(1) and q(2) are distinct primes): If D is an element of Z[Z/q(1)q(2)Z], then chi (D) is an element of Z for every character chi : Z/q(1)q(2)Z -%26gt; C-x if and only if psi (D) is an element of Z(p) for every prime p and every character psi: Z/q(1)q(2)Z -%26gt; Q(p)(xi(n)).

• 出版日期2013-1-1