Abstract Algebra, Lec 37: Galois Theory: Definitions, Two Examples, Insolvability of a Quintic

(0:00) Field Theory and Galois Theory, Part 10.
(0:48) Old photos of me.
(2:38) Definition of a Galois group of a field extension of a field.
(5:46) Outline of proof that Gal(E/F) is a subgroup of Aut(E) (two-step subgroup test).
(9:33) Alternative notation.
(9:54) Fixed field of a subgroup of the Galois group and trivial examples.
(12:05) Example 1: Find all the subfields of Q(sqrt(3),sqrt(5)) (and make a lattice of subfields) that contain Q by finding the Galois group of Q(sqrt(3),sqrt(5)) over Q.
(15:33) What are the elements of the Galois group?
(16:56) If E is an extension field of Q, then any automorphism of E fixes Q (and the idea of the proof).
(22:45) How do elements of this Galois group act on sqrt(3), sqrt(5) (and sqrt(15))? (Also write elements of Q(sqrt(3),sqrt(5)) as linear combinations of the basis elements {1, sqrt(3),sqrt(5),sqrt(15)}).
(28:47) Write the possible general formulas and use Mathematica to verify that they are operation-preserving (and show one example that is not operation-preserving).
(35:07) Write down the Galois group for Q(sqrt(3),sqrt(5)) over Q as a 4 element group (group of order 4) isomorphic to the external direct product of Z2 with Z2.
(37:03) Subgroup lattice of the Galois group, fixed fields of the subgroup of the Galois group, subfield lattice, indices, and degrees (Fundamental Theorem of Galois Theory).
(43:41) Example 2: Consider Q(ω,cuberoot(2)), where ω is a (complex) primitive cube root of unity 1 and cuberoot(2) is the real cube root of 2.
(48:10) Start to make the lattice diagram (tower of fields) of subfields of this extension.
(49:12) Draw cuberoot(2), ω*cuberoot(2), and ω^2*cuberoot(2) in the complex plane.
(50:24) The Fundamental Theorem of Galois Theory allows us to conclude Galois group has order 6 so is isomorphic to either Z6 or S3 (and S3 is isomorphic to D3).
(51:57) Facts about the zeros and their minimal polynomials and how the zeros get permuted under elements of the Galois group, which allows you to determine possibilities for elements of the Galois group.
(56:50) Formulas, in terms of linear combinations, for two "generators" of the Galois group, called α and β.
(1:01:04) Orders of α and β in the Galois group. The Galois group is non-Abelian and is isomorphic to S3, the symmetric group on 3 objects (permutation group).
(1:02:41) Write alpha and beta as permutations in cycle notation (also represent in the complex plane and realize that alpha is really complex conjugation).
(1:04:57) A quintic that is not solvable by radicals (the quintic is g(x) = 3x^5 - 15x + 15). The Galois group is isomorphic to S5, which is not a solvable group.

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[billkinneymath]

Typos at 53:20 are: 2^(1/3), omega*2^(1/3), and omega^2*2^(1/3) are zeros of g(x)=x^3-2 (NOT g(x)=x^3-1)

[jimzhang]

Thank you so much for the video, very clear explanation of Galios Theory.

[elizabethmrema]

I never attended Galois theory lecture. Thank you for the nice clarification as it opened up my mind in understanding Galois extension. Can you please share with me the name of the book you are using if you don't mind! Thank you