# Galois Theory

October 20, 2011 2 Comments

First off, I’m spreading posts out from now on. I just don’t have the time to push out this much math when I’m supposed to be doing other math. So what is Galois Theory? The idea is to look at fields and and how fields contain each other, and relate them to groups and how groups contain each other. This will (though it may seem odd) depend largely on roots of polynomials. This is good though, because it will allow us to answer questions about polynomials (like the existence of a quintic equation). There are a few things you should be comfortable with to really attack Galois Theory

- Fields, field extensions
- Considering one field as a vector space over another
- Groups, normal subgroups (I’ll remind you about these when the time comes)
- Complex numbers
- Being badass (this one is crucial)

**algebraically closed field**is one in which every polynomial with can be factored into linear terms. If you know the fundamental theorem of algebra, another way to state it is that is algebraically closed. An example of a field that is not algebraically closed is . The polynomial cannot be factored over .

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