The Structure of Fields: What is a field and a connection between groups and fields

This video is primarily meant to help develop some ideas around the structure of fields and a connection between groups and fields (which will allow me to create more abstract algebra videos in the future! 😀😅🤓)

00:00 Intro
01:04 What is a Field?

Here we give the definition of a field in math and break down the different components, ring, commutative ring, with unity, and units all step by step to give us a good foundation for what we are talking about when we talk about fields in this video.

03:46 Examples and Non-Examples of Fields

As the title of this section suggests, we give some examples of fields (the rationals, the reals, and the complex numbers) as well as some non-examples with the help of particular integers and matrices with different entries.

05:50 Polynomial Factorization and Extension Fields
Here we give the algebraic view of polynomials and their mathematical habitat, polynomial rings. After which we ask, " Given a polynomial p(x) in the polynomial ring formed by K and a single variable/symbol, can we factor p(x) into linear factors only using the elements of K as coefficients, or do we need something more in our field?" This question motivates the idea of an extension field and also produces a field and polynomial factorization diagram that begins to move us toward the connection between groups and fields.

12:52 Connecting Groups and Fields
Continuing on with the diagram from the last section, we start to look at how the polynomial factorization and how "free" we were to mess with how we extended the fields without permuting the factorization in each case. This question allows us to form Galois groups, which are groups of transformations, and gives us an example of the Galois Correspondence in action.

An Exercise that is also a note:
When using polynomial factorization as the drive for the Galois Correspondence we sometimes miss certain field extensions and groups that correspond to one another. In the example from this video, we actually miss one such field/group pair. What is that field group pair? This field/group pair gives some motivation for another, "more complete" idea for how to formulate the Galois correspondence.
(Hint: The answer is not in the video but, I guess you could say it is on the boundary of the video.)

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WHAT GEAR I USED FOR THIS VIDEO:

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Subscriber Count as of release: 1,631

#CHALK #GaloisTheory #Fields

[mooglg]

Showing the correspondence that way was incredibly enlightening and shows exactly why there's that "flipping" when chaining the automorphism groups. This is awesome! Thanks!

[sadface]

Seen some ambitious cross-overs but groups and fields must be in the top 10

[viniciusalmeida]

This channel should have waaaay more subscribers! Thanks for the great job man!

[tomijantunen]

Like I said in another video, I'm in the beginning stages of my masters in Finland. This is exactly the stuff we are learning about at the moment. Having some problems with the material, because it turns out in this school they went a bit further into abstract algebra than we did in our school. Holding on though! Really enjoyed this video, thank you for making it. Helped me learn about the big picture.

[Santi._.]

Dope music at the end, send links please.

[SpiritVector]

This is incredible buddy, holy jeez! No one on YouTube is doing this right now so keep it up!

[usmleck]

Where are you, what are you doing now in December, please keep us updated about your PhD? What’s the plan for 2021?