- #1

PsychonautQQ

- 784

- 10

## Homework Statement

If Char(K) = 0 and [L:K]=2, is L:K a galois extension?

## Homework Equations

## The Attempt at a Solution

My gut is saying yes because if [L:K]=2 then it seems that any polynomial in K[x] with a root in L should split in L[x]. Something about how some hypothetical minimal polynomial of some element m call it m(x) where L[x] is isomorphic to the quotient field K[x]/m(x), then K(u) is isomorphic to L where given the conditions that [L:K] =2 then deg(m(x)) = 2 and so if L has one of the roots of m(x) then that means that m(x) factors linearly because it only has degree 2 and thus all the roots are in L thus the extension is normal. Am I leaving anything important out?