Visual Group Theory, Lecture 6.1: Fields and their extensions

Visual Group Theory, Lecture 6.1: Fiends and their extensions

This series of lectures is about Galois theory, which was invented by a French mathematician who tragically died in a dual at the age of 20. He invented the concept of a group to prove that there was no formula for solving degree-5 polynomials. Galois theory involves an algebraic object called a field, which is a set F endowed with two binary operations, addition and multiplication with the standard distributive law. Formally, this means that (F,+) and (F-{0},*) must both be abelian groups. Common examples of fields include the rationals, reals, complex numbers, and Z_p for prime p. In this lecture, we examine what happens when we begin with the rational numbers, and the "throw in" roots of polynomials to generate bigger fields called "extensions".

[anjummuneer]

I was having problem in understanding this concept, but after watching this lecture, I truly appreciate your effort.

[howmathematicianscreatemat]

I must deeply thank you for this!! This is really what I have missed so much throughout my studying for Bachelor-mathematics!! You help us so much and really enable us to understand these thrilling things intuitively! Thanks!! :-))

[stuartneil]

I’ve been pushing and pulling my brain through a book in Galois Theory by Emile Artin. Your video is so much more concise and helps a lot. Thanks.

[pss]

Best channel for Galois theory. Thank God I found this!

[liamgauvreau]

For the homework question near the end, I believe you can adjoin the roots of either x^2-3 or x^2-2 and get the same field extension.

[namirahfatmanissa]

This video is very helpful! Thank you for being very detailed and the transition from the simplest to the more advanced examples is very enlightening

[waylonjepsen]

I have to say your lectures are very clear. This is greatly appreciated.

[ali]

Excellent. So nice and explicitly was presented. If the thump up did not have a limit, I would have stay all day and all night long up to click and click on it and still not enough. Thank you

[rajendramisir]

The format and methods along with the sound explanation of these lectures will work well for an Online degree program in applied and computational mathematics. Thank you for your hard work and sharing it online.

[zofiawajda]

Honestly, the first field theory lectures I came across that helped me get good intuitive understnading

[kvasios]

Oh man, fantastic course. It all starts coming together.

[madrid]

Thank you Professor. Please post more. These are great to review concepts.

[ghazalfaris]

this is by far the Best introduction to Fields, & group theory as a whole, with many Takeaways to learn from
Thank you so much Professor,
Field
F Abelian Group
F \{0} (without 0 ) closed under multiplication
3. Distributive law holds: a*(b+c) = a*b +a*c

Field {is a Ring} where you can do basic Arithmetic

sets; not fields: N Z, Zn (n composite)
Fields: Q R C, Zp (prime)


Extension
coeffs of linear equationa a__+b___ are the Extension Q(, )
i.e. a*sqrt(2) + b*

19:00 supporse have a Vector Space, Spanned by
V1, V2, V3, V4
any element in little x written as
a*v1 + bv2 cv3 + dv4
which forms a 4D vector space

like in group & subgroup, we arrange Fields & subfields..

graph: label each extension as Degree

Q. Abelian Group does not have Division (/) [only in Hilbert's]
it has only + * - } as 3 operations, 1-to-1 & onto, correct?

[luisianypomalesnegron]

I love your explanation! You did it good! Keep up the good work! Hope you make more videos about visual group theory!

[prfontaine]

Some parts of the presentation are factually correct but potentially misleading I think. When one adds (adjoins) an element x to a field k, forming k(x), we obtain not only {a + b.x | a, b E k } but the bigger set {P(x) | P E k[X] } where k[X] denotes the set of all polynoms of variable X with factors in k. Notice that k(x) is not a field in general (this should be said explicitly). On revenge what can be proved is that is that k(x) is a vector space over k of finite dimension if x is algebraic over k. This is why your examples work: you always adjoin to Q values that are algebraic over Q and this is why you can obtain a finite basis for the vector space k(x).

I think that the definition of k(x) should be separeted from the extension field notion as k(x) is not always a field.

[streetwear]

As I recall, a field are given a set F with any to arbitrary binary operators + and * (they are just symbols representing some binary operator), with e_+, e_* are identity elements satisfying

1. (F, +) is an abelian group and
2. (F\{e_+}, *) is an abelian group.
3. For a, b, c in F, a*(b+c) = a*b+a*c.

[rachelnanshija]

Hey friend. It's probably a not insignificant amount of work to do so, but could you organize your videos into playlists? I like your visual group theory lectures and I would like to be able to binge watch them easily or find them quickly by saving the playlist. It would be a really helpful study tool, as well as make sharing an entire series of lectures with friends easier.

[dipeshbarman]

Really awesome.Nicely explained.Thank you sir

[jhanschoo]

Ah I understand the link between the last two questions:




Yeah, it has structure V_4; not just a subgroup lattice V_4. The second last qn.'s answer has that adjoining x^2-3 or x^2-2 works to make Q(sqrt(6)) into Q(sqrt(2), sqrt(3)), but moreover, as in V_4, you can adjoin x^2-6 to either Q(sqrt(2)) or Q(sqrt(3)) to make either Q(sqrt(2), sqrt(3)), similar to V_4.

[jehushaphat]

mistake on 16^2...nevertheless it doesn't take anything away from your lesson. Very helpful..been trying to understand field extensions and no one made it as clear as you did. Thanks!