Visual Group Theory, Lecture 6.4: Galois groups

Visual Group Theory, Lecture 6.4: Galois groups

The Galois group Gal(f(x)) of a polynomial f(x) is the automorphism group of its splitting field. The degree of a chain of field extensions satisfies a "tower law", analogous to the tower law for the index of a chain of subgroups. This hints at a deep connection between subfields of a splitting field and subgroups of its Galois group, which we will uncover soon. Also in this lecture, we learn how every finite degree extension of the rationals Q is "simple", which means that it is generated by a single element that we call a "primitive element".

[brianbecsi]

I like that there are like 24k views and only 270 likes which means that this is like some of the most helpful content out there. I've got my final today LFG!

[speedbird]

Very nice visual proof of the tower law!

[thomashoffmann]

1/(sqrt(2) + sqrt(3)) did the trick for the last question. You can derive sqrt(2) and sqrt(3) easily with that primitive element.

[leostokes]

Is Joe Brawley still at Clemson? I went to N. C. State with him in the 60s. A long time ago! We tried to learn Galois theory there from Nathan Jacobson's books. I learned it incompletely at U of Illinois. Thanks for your lessons. The theory is now revealed. Bravo.

[chetedoherty]

2 questions:

1) In the previous video at the very end, it determined that zeta*cube root 2 has minimal polynomial (x^3)-2 over Q and that [Q(zeta*cube root 2):Q] = 3. Here we find [Q(alpha):Q]=6. This feels contradictory to me.

2) In the final exercise, what is the minimal polynomial? alpha^4 should give us an element in Q, but I calculate it as 49 + 20*sqrt 6

[howmathematicianscreatemat]

May I give a small reminder that alpha^4 = 18*2^(1/3) and not -18*2^(1/3)

[MostlyIC]

when you say the only automorphism of Q(sqrt(2)) is isomorohic to C2 aren't you assuming the only automorphism of Q itself is the identity map ?, isn't that a pretty big assumption worthy of explanation ?

[samtux]

Great lecture series! Deep understanding of this complex topic and nice visual examples.

But... So many arithmetic errors... 25:55 alpha^4 should be "+18*2^(1/3)" not "-18*2^(1/3)" .
Earlier : [254 =256-2 = 16^2-2 =] (16+2^0.5)(16-2^0.5) = (??) 194

[hyperduality]

Conjugate root theorem implies duality, roots come in pairs!
Conjugate = Duality!
"Sith lords come in pairs (duals)" -- Obi Wan Kenobi.
"Always two there are" -- Yoda.
SINE is dual to COSINE -- the word "co" means mutual and implies duality.
SINH is dual to COSH -- hyperbolic functions.
Injective is dual to surjective synthesizes bijective or isomorphism.
Conjugate pairs = automorphism!
Sub groups are dual to sub fields -- the Galois correspondence.