Field Theory and Galois Theory, Part 9. (0:00) Field theory counterpart (for degrees of fields extensions) of Lagrange’s Theorem and how to find a basis for a secondary extension over the original base field. (6:09) Primitive element theorem for fields of characteristic zero. (7:47) Transitivity of algebraic extensions. (8:43) Finite field (Galois field) classification and description of subgroups. (13:18) We will be introducing the basics of Galois theory here at the end of Lecture 36B and all of Lecture 37 (both parts A and B). (14:29) The big picture of what Galois theory is about (and a reminder about what groups of automorphisms are). (19:31) Galois’ goal was to prove that the general quintic (fifth degree) equations are not solvable by radicals. (23:05) The Galois group Gal(Q(sqrt(3))/Q) of Q(sqrt(3)) over Q.

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