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Field orderable iff sym matrixes diagonalizable over closure

Source: Miklós Schweitzer 2017, problem 2

January 13, 2018

linear algebramatrixabstract algebrafield

Problem Statement

Prove that a field

K

can be ordered if and only if every

A\in M_n(K)

symmetric matrix can be diagonalized over the algebraic closure of

K

. (In other words, for all

n\in\mathbb{N}

and all

A\in M_n(K)

, there exists an

S\in GL_n(\overline{K})

for which

S^{-1}AS

is diagonal.)