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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

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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!!

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    $\begingroup$ Hi: for future reference it would probably be better to say "I can't follow the argument" rather than you are "unable to think about it." Being unable to think about something would indeed be an extreme handicap, but it sounds a lot more silly in English than what I think you meant ("I don't understand the step in the argument") $\endgroup$
    – rschwieb
    Apr 1, 2020 at 12:45
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    $\begingroup$ @rschwieb Ok , I will keep that in mind!! $\endgroup$
    – user775699
    Apr 1, 2020 at 12:47

1 Answer 1

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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 $u \in H'$.

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