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Let F be a local number field. We give necessary and sufficient condi­ tions for a monic polynomial p(x) over F to be the characteristic poly­ nomial of a symmetric matrix over F. (A matrix "over F" is a matrix whose elements lie in F.) In the process we obtain a lifting formula for the Hilbert symbol. Proofs will appear elsewhere ((2), (3), (4)). Notation and terminology. We denote the transpose, determinant, and dimension of a square matrix X by X\ \X\ and dim X respectively. Let p(x) = qi(x)q 2(x)- • • be the prime decomposition of p(x) in F(x). Define Gt = F(x)/(qt(x)) and se = Gx © G2 0 • • •.

Symmetric matrices, characteristic polynomials, and Hilbert symbols over local number fields