I am self studying Fields and Galois Theory from Tomas Hungerford and I am unable to think about an argument in lemma in Fields and Galois Theory
Adding it's image ->
I have question in only Line 2 of proof of part (ii) of this lemma -> I am not able to deduce why $\sigma^{-1}\tau\sigma(u) $= u must hold?
I tried using information in line 1 of lemma(2) but can't deduce?
Can anyone please help!!
The field $H'$ is by definition the fixed field of $H$. This means that every element of $H'$ is fixed under every automorphism that lies in $H$, i.e. $\varphi(h') = h'$ for all $\varphi \in H$ and $h' \in H'$. Now we know that $\sigma^{-1} \tau \sigma \in H$ (line 1) and hence $\sigma^{-1} \tau \sigma(u) = u$ for every element $u \in H'$.