Visual Group Theory Lecture 61 Fields and their extensions

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Visual Group Theory, Lecture 6.1: Fiends and their extensions • This series of lectures is about Galois theory, which was invented by a French mathematician who tragically died in a dual at the age of 20. He invented the concept of a group to prove that there was no formula for solving degree-5 polynomials. Galois theory involves an algebraic object called a field, which is a set F endowed with two binary operations, addition and multiplication with the standard distributive law. Formally, this means that (F,+) and (F-{0},*) must both be abelian groups. Common examples of fields include the rationals, reals, complex numbers, and Z_p for prime p. In this lecture, we examine what happens when we begin with the rational numbers, and the throw in roots of polynomials to generate bigger fields called extensions . • Course webpage (with lecture notes, HW, etc.): http://www.math.clemson.edu/~macaule/...

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