次世代の哲学を育むスレ Part 10

148 ：考える名無しさん：2020/08/05(水) 17:30:46 ID:0.net

This name is chosen because in the case where the σ’s are automorphisms and
σ1 is the identity, i.e., σ1(x) = x, we have σi(x) = x for a fixed point.

Lemma. The set of fixed points of E is a subfield of E. We shall call this subfield
the fixed field. For if a and b are fixed points, then σi (a + b) = σ1 (a) + σi (b)
= σj (a) + σj (b) = σj (a + b) and σj (a 。b) = σi(a) 。σi(b) = σj(a) 。σj(b)
= σj(a 。b). Finally from σi(a) = σj (a) we have (σj(a))–1 = (σi(a))–1 = σi(a–1) = σj(a–1).

Thus, the sum and product of two fixed points is a fixed point, and the inverse
of a fixed point is a fixed point. Clearly, the negative of a fixed point is a fixed point.