Galois theory: Examples of Galois extensions

This lecture is part of an online graduate course on Galois theory.
We give several examples of Galois extensions, and work out the correspondence between subfields and subgroups explicitly.

[@brunodular]

Thank you so much for all these videos!!!

[@Brien831]

I am the 1337th viewer and the 69th like. I am so gonna ace my algebra exams haha.

[@Brien831]

I failed

[@vs-cw1wc]

@@Brien831 don't give up bro

[@realwermos]

I feel you, I'm on my way to the same fate 😢😭

[@jacobsmith7285]

I have no idea what any of this means but I wanna know more!

[@vs-cw1wc]

Is "lattice" a mathematical terminology that can be precisely defined?

[@nnaammuuss]

Yes. This case of lattice refers to a partially ordered set.. with exactly what properties I don't remember.. they should definitely have joins and meets (a join of two elements a, b is the _maximum_ element (which is hypothesized to exist) of the subset { c : c ≤ a, c ≤ b }-that's the _intersection_ in case of a powerset; the meet is the dual concept) . Some properties such as associativity, commutativity, idempotence of joins and meets I think just follow. What doesn't follow in general is the distributivity w.r.t. each other and isn't hypothesized. Thus, the main theorem of galois theory needs a little more work showing that in the correspondence, these operations (dualized) correspond too-but that's not hard.
There's of course another notion which also goes by the name of lattice: viz. a finitely generated Z-submodule (additive subgroup) of some R^n (R meaning the reals). This is not the one he was talking about, but this too shows up in these contexts, eg. when you're trying to compute a galois group (starting from an equation), dealing with factorization of rational polynomials, and in particular, the LLL algorithm. 👍

[@nnaammuuss]

Sorry, if two lattices (in the first sense) are shown to be isomorphic as posets, the join and meet operations obviously correspond. No extra work required.

[@migarsormrapophis2755]

ye

[@benhsu42]

11 minute mark: why is F8 not a subfield of F16?

[@jacobfertleman1980]

Didnu figure it out?

[@benhsu42]

@@jacobfertleman1980 not yet

[@abderrahmaneprofmaths659]

If K is a subfield of L, then L is a vector space over K.
In the mentioned example, any (finite dimensional) vector space over F_8 must have cardinality 2^{3k}.
In general F_p^{m} is a subfield of F_p^{n} iff m|n

[@benhsu42]

@@abderrahmaneprofmaths659 Thank you very much!