302.S2a: Field Extensions and Polynomial Roots

[PunmasterSTP]

I was following along through the MATH 302 Abstract Algebra II playlist and got a bit worried it ended, but I'm glad to see more videos in this Abstract Algebra II playlist. Thanks again for making such high quality educational material and sharing it for free!

[ouafieddinenaciri]

Hi
There is a redundance and a missing :
On the fourth column we already have 1+t+t^3
On the seventh column we shouldn't have the same thing but t+t^2+t^3 and t+t^2+t^4 and so on ...
Am I wrong ?
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[vanessamichaels]

youre the best! I knew when I saw that it was one of your videos that I would get it.

[dmj]

At 4:00, why can't we think of F as being a subset of E? I thought it was customary to consider the base field as a subset of the extended field.

[IanFarias]

Thank you very much, this is the kind of insight I was hoping to get from field extensions! ^^

[pennerzhang]

Thank you Matthew! This is the best video I found in youtube for this concept.

[Mrpallekuling]

02:37 This polynomial and the related splitting field of t^4-2 over Q is discussed in detail (4 pages) in Ch 13 "A Worked Example" in Ian Stewart: Galois Theory.
That chapter is very instructive. Stewart says "it is a favorite of writers on Galois theory" and "A simpler example would be too small to illustrate the theory adequately, and anything more complicated would be unwiedly".
At the time of writing, Edition 4 of Stewart's book is available online (the latest edition is edition 5).

[nordgothica]

I do not understand how we find the elements of the that quotient field by setting the polynomial equal to 0. Can anyone explain to me what it is we are doing here, or the intuition behind it?