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Suppose $K$ is a prime subfield of $E$, then if $\phi$ is an automorphism from $E$ to $E$, we have for all $x \in K$, $\phi(x) = x$.

I feel like this is just the definition of a field automorphism, but my book says this should be proven as an exercise.

quid
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Ninosław Brzostowiecki
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  • You're confusing field automorphism and K-automorphism of a field extension of $K$. – Bernard Apr 25 '16 at 22:46
  • @Bernard That is not the issue here. Notice that $K$ is the prime subfield. – Matt Samuel Apr 25 '16 at 23:06
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    I did notice it, and it seems to me the O.P. didn't know the prime subfield can be defined as the smallest subfield of $E$. – Bernard Apr 25 '16 at 23:16
  • For anyone looking to undo my retagging, know that I disagree that this is not ring theory and I will flag if the ring-theory tag is removed. This is not Galois theory, this is a statement about fields as rings. – Matt Samuel Jul 15 '20 at 21:42
  • @MattSamuel You’re right. This isn’t [tag:galois-theory]. It isn’t [tag:ring-theory] either. Do you care to explain what’s your agenda here? This is extremely weird … – k.stm Jul 15 '20 at 21:46
  • @k.stm "This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory." – Matt Samuel Jul 15 '20 at 21:55
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    @MattSamuel You forgot to tag this question [tag:group-theory], [tag:ring-homomorphism] and [tag:extension-field] then. Also don’t forget to add [tag:automorphism-group] and [tag:commutative-algebra]. But please do so only in three years, so that an answered question gets bumped again at a random time for no sensible reason whatsoever. – k.stm Jul 15 '20 at 22:00
  • @k.stm I answered a question tagged ring-theory but it's about formal power series rings. Are you going to retag that as formal-power-series? – Matt Samuel Jul 15 '20 at 22:07
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    @MattSamuel Nah, I wouldn’t even bother with unneccesarily retagging old questions. That’d be weird. But you know what? You do your thing, man. Apparently, having these old questions being tagged “ring-theory” (even though they clearly aren’t to anyone taking the term seriously) is for some secret reason very important to you. I had been bothered by your retags because they bump up zombies, which annoys me, and then I got curious as to why someone would do this. But whatever, man, you do your thing. Go wild. – k.stm Jul 15 '20 at 22:14
  • @ArcticChar I have flagged the post as promised. Please stop removing the ring-theory tag. – Matt Samuel Jul 16 '20 at 10:19
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    I changed the tags again; anyone that wants to continue this debate can find me in chat. I might be willing to change my mind but any direct retagging is forbidden now until further notice. – quid Jul 20 '20 at 12:48

1 Answers1

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This is because the prime subfield is generated as a field by $1$. Since you have no choice but to send $1$ to itself, the prime subfield remains fixed as well.

Matt Samuel
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  • I see, that makes sense since an automorphism is by definition an isomorphism. – Ninosław Brzostowiecki Apr 26 '16 at 13:22
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    Please take care, @Matt, not to retag with inappropriate tags, the posts you've answered. – amWhy Jul 15 '20 at 00:10
  • Why are the tags inappropriate? At its heart, this question is really about the fact that any ring morphism $R \to S$ is a morphism of $\mathbb{Z}$-algebras. I don't think "ring theory" is such an inappropiate tag. – Alex Wertheim Jul 15 '20 at 02:37
  • @AlexWertheim $ℤ$ is commutative. This must be a question about [tag:commutative-algebra]! Please note that both the question and the answer would make perfect sense even if the world had never known what a ring was. To tag this [tag:ring-theory] after four years is very weird. I want to know what’s behind this, to be frank. – k.stm Jul 15 '20 at 21:54
  • @k.stm: my reasons for thinking the ''ring theory'' tag applies are not just word association, like what you suggest. My feeling is that while the question asked specifically about fields, the answer should be cast at the level of generality of rings. That makes the ''ring-theory'' tag appropriate to me. If you find it unpleasant that an old post has been bumped over a trivial edit, that's fine, but that's a different issue. (Maybe MSE should not bump posts over tag edits or something?) – Alex Wertheim Jul 16 '20 at 00:45
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    @AlexWertheim That argument would make sense if the answer had actually been cast at the level of generality of rings. But it has not. The purpose of tags is to group together contents within similar contexts. Since neither the question nor the answer has been put into context of ring theory, the tag is inappropriate. – k.stm Jul 16 '20 at 05:34