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Since we start with field theory, we would have to jump into the middle of most graduate algebra textbooks. This can make reading the text difficult by not knowing what the author did before the field theory chapters. Therefore, a book devoted to field theory is desirable for us as a text. While there are a number of field theory books around, most of these were less complete than I wanted. JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser.

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.

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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.

Visual Group Theory, Lecture 6.1: Fields and their extensions

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