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The group of field automorphisms
Recall that a field automorphism is a function that “respects all of the properties of fields.” For instance, if is a field automorphism, then , , , and . I did leave out one rule of note . Why did I...
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Automorphisms of Q
We’ve talked a bit about automorphisms, but we haven’t seen very much of them, so I wanted to do an example. What is (the group of automorphisms of the field of rational numbers)? It definitely...
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Fixed fields
If I have a field and a field automorphism , I can ask about the fixed points of . That is, all of the such that . Notice that if fixes and , then and , so fixes and . In fact, the set of fixed points...
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Galois Group
Last time we started with a subgroup of the automorphism group of a field , and asked about it’s fixed field, the field of elements which are fixed by every automorphism in the subgroup. We noticed...
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Galois Connection
So we mentioned two extremely important concepts to Galois theory already. The Galois group of a field extension of a field . This is the group of automorphisms of that fix . We’ll denote this with The...
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Galois is finished
Over the past month or so, two issues have arisen that have prevented me from posting regularly. The first is that free time has been scarce. The second is that I have grown slightly bored of Galois...
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Global choice and algebraic closures
It was pointed out to me today that I glazed over a set-theoretic point in my proof that every field has an algebraic closure. We appealed to Zorn’s lemma, which says: Given a partially ordered set ,...
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Applying Yoneda’s Lemma
Let’s write down Yoneda’s lemma one more time. For a locally small category, and functors from that category into , . One instance of Yoneda’s lemma is to take the second functor to be . Then we have a...
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A proof via Lagrange’s theorem
Let’s assume for the sake of contradiction that there are finitely many primes. Let denote the biggest prime. Since , it cannot be prime. It must be divisible by some other prime . Another way to say...
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KnowMSG v1.4
First things first, for those of you actually reading on my blog, and not via an RSS feed, you probably noticed I changed the theme. I’m not sure if I’ll stick with it, but for now, I like it. Now that...
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