Abstract Algebra, Lec 38: Galois Group Example, Grad School Topics, Review for Final Exam

Find the Galois group of f(x) = x^4 + 1 over the rationals Q. Grad school topics include: category theory, homological algebra, commutative algebra, advanced linear algebra. Final Exam Review. "Contemporary Abstract Algebra", by Joe Gallian: https://amzn.to/2ZqLc1J
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Abstract Algebra, Lecture 38. Field Theory and Galois Theory, Part 11

(0:00) The last day of class (and lecture plan).
(1:02) Review fundamental definitions: group of automorphisms Aut(E), Galois group Gal(E/F),
fixed field (and verbal reminder of what the Fundamental Theorem of Galois theory is about).
(6:45) Example: find the Galois group of f(x) = x^4 + 1 over the rationals Q (and define the Galois group of f(x) over Q as the Galois group of its splitting field over Q).
(8:20) Think about complex arithmetic to find the splitting field of f(x) = x^4 + 1 over Q to be Q(omega), where omega = 1/sqrt(2) + i/sqrt(2)) (omega is a fourth root of -1 and, in fact, a primitive 8th root of unity (of 1)).
(18:47) Goals: for the Galois group Gal(Q(omega)/Q) we want to: 1) determine its order, 2) determine its isomorphism class, 3) determine its elements, 4) determine the fixed fields of its subgroups, and 5) draw lattice diagrams of both the Galois group and the field extension (make a tower of fields).
(19:19) Any automorphism of a field extension that contains Q automatically fixes elements of Q.
(20:42) The identity mapping is in it.
(21:13) Automorphisms of E = Q(omega) will permutes the roots (zeros) of any polynomial in Q[x] with roots in E.
(24:53) Not just any permutation of the roots will give an automorphism.
(25:48) Determine the four elements of the Galois group by seeing that they are completely determined by how they map omega.
(35:50) Gal(E/Q) is a group of order 4 and is isomorphic to the external direct product Z2 + Z2 (could also check that these automorphisms are operation-preserving by writing the elements as a linear combination a + a*omega + b*omega^2 + c*omega^3 and coming up with formulas for the automorphisms).
(40:57) Subgroup lattice diagram for Galois group.
(41:59) Lattice of subfields (which are fixed fields) of the extension.
(43:11) There are quintics that are not solvable by radicals and "Solve" in Mathematica won't help (though it does give "nasty" formulas for general cubics and quartics).
(44:01) Summary of advanced topics covered in graduate school: category theory, homological algebra, commutative algebra, advanced linear algebra.
(48:46) Review for the Final Exam of Abstract Algebra.

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