- #1

Delurker

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*A Book of Abstract Algebra*, 2nd ed, by Pinter and I have two questions. First, the author says to consider the situation where ##K## and ##K'## are finite extensions of ##F##, and furthermore that ##K## and ##K'## have a common extension ##E##. Then he goes on to prove that if ##K## is a splitting field of ##F## and ##h:K\rightarrow K'## is an ##F##-fixing isomorphism, then ##h[K]\subseteq K##. (Pinter uses the term "root field" instead of splitting field and I've changed it to the more standard term.) Then comes the theorem:

Let ##K## and ##K'## be finite extensions of ##F##. Assume ##K## is the splitting field of some polynomial over ##F##. If ##h:K\rightarrow K'## is an isomorphism which fixes ##F##, then ##K=K'##.

Proof: From Theorem 2 [

*this is the primitive element theorem; we're assuming fields have characteristic zero*], ##K## and ##K'## are simple extensions of ##F##, say ##K=F(a)## and ##K'=F(b)##. Then ##E=F(a,b)## is a common extension of ##K## and ##K'##. By the comments preceding this theorem, ##h## maps every element of ##K## to an element of ##K'## [

*this is trivially true so the last prime must be a typo*]; hence ##K'\subseteq K##. Since the same argument may be carried out for ##h^{-1}##, we also have ##K\subseteq K'##. QED

If my understanding is correct, then we have to assume that ##K## and ##K'## are contained in a common extension for the theorem to be true. I would have just accepted the fact that Pinter didn't explicitly mention this in the theorem because he did a page ago, but the proof implies that we can always create an extension, which isn't true (it's not possible if ##K## and ##K'## are isomorphic but not equal splitting fields over F, for example). As the theorem stands, without explicitly assuming a common extension, it seems that letting ##F=\mathbb{Q}, K=\mathbb{Q}(\sqrt{2})##, and ##K'=\mathbb{Q}[x]/\langle x^2-2\rangle## is a counterexample. Does that sound about right? It would clear up another confusion I had when I learned (not too well, obviously) from Fraleigh's book years ago. I didn't quite appreciate why he made sure to say that we're assuming that all work is being done in one fixed algebraic closure before developing much theory. Is it pretty much to avoid technical issues like this?

The second question is actually about the first edition of the book. He states a theorem that if ##I\subseteq E\subseteq K## with ##K## a finite extension of ##E## and ##E## a finite extension of ##I##, and ##K## is a splitting field over ##E## and ##E## is a splitting field over ##I##, then ##K## is a splitting field over ##I##. I don't think this is true. I tried to find some counterexamples but they were too simple and didn't work. I'm not sure if I should spend more time on it considering that there is an obvious gap in his proof and the theorem doesn't appear in the second edition (and it would make later results considerably easier to prove).