Readings | Number Theory II: Class Field Theory | Mathematics | MIT OpenCourseWare
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Now the crux of The Field Story is the construction of such splitting fields. This construction is analogous to building a tower from the ground up - one floor at at time. I'm doing a lot of hand-waving here, but we eventually obtain a tower of fields which looks something like this:.
And that group is our next topic of discussion. Do you remember those verb commercials from the early to mid s? You know, the ones full of hyper-happy, active kids encouraging you to do some physical activity? Just Google it , you'll see.
For some reason I do. The tag-line was. And in the context of mathematics, groups are very much like verbs!
The Basic Idea
They do stuff. In particular, a group acts on a set by shuffling elements around. We can gather a lot of information about the group and about the set it acts on via this action. So if you ever want to create a TV commercial about mathematical groups,.
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But how does this relate to Galois theory? Recall a field automorphism is just an isomorphism from a field to itself. This group is isomorphic to i.
And with that, we end our bird's-eye view of a course in Galois Theorey. Kudos if you made it through the whole thing! I hope you've gotten a little taste of what Galois Theory is about. But this is really just the tip of the iceberg.
Field and Galois Theory
For further reading, I highly recommend Basic Abstract Algebra 2 ed. This book is very approachable at the undergraduate level. And of course, Abstract Algebra 3 ed. Many thanks to a reader for pointing this out. September July May March January November October