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!
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).
I think we can actually (why complicate matters?). For example, Wolfram Mathworld simply says: "A field K is said to be an extension field (or field extension, or extension), denoted K/F, of a field F if F is a subfield of K"
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?
3 years late, but for other commenters: what is being done here is essentially modulating up and until exactly the polynomial in the quotient group. The reason is that now the polynomial is exactly 0 when applied to t. Think of it as "what group can we construct so some integer B is zero?"; the easiest solution is to make a group Zb - since it modulates back to zero upon reaching b. Filling in t for t^3 + t + 1 is itself, which is declared zero in the extension. It sounds pretty stupid, but it clearly works.
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 ? Friendly
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!
Thank you Matthew! This is the best video I found in youtube for this concept.
youre the best! I knew when I saw that it was one of your videos that I would get it.
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).
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.
I think we can actually (why complicate matters?). For example, Wolfram Mathworld simply says: "A field K is said to be an extension field (or field extension, or extension), denoted K/F, of a field F if F is a subfield of K"
Thank you very much, this is the kind of insight I was hoping to get from field extensions! ^^
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?
3 years late, but for other commenters: what is being done here is essentially modulating up and until exactly the polynomial in the quotient group. The reason is that now the polynomial is exactly 0 when applied to t. Think of it as "what group can we construct so some integer B is zero?"; the easiest solution is to make a group Zb - since it modulates back to zero upon reaching b.
Filling in t for t^3 + t + 1 is itself, which is declared zero in the extension. It sounds pretty stupid, but it clearly works.
The strategy is to find an extension field E where the polynomial has a root.
One way to do this is to create a quotient field E=F[t]/
What’s a monomorphism here?
I might be wrong, but in this context I think it means one-to-one.
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 ?
Friendly
I think you're right; good catch!
How is this useful for say a scientist who needs to find the root of a polynomial? Why would the be dissatisfied with a complex root?
nonono，i cannot understand😭😭😭