The Third Pillar Of Algebra: Fields | Abstract Algebra | Field Theory | Dogmathic

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What is a field, really? We start from scratch with a set F and two operations, + and ·, and then write out the axioms instead of hiding them behind one sentence.
First, (F,+) must be an abelian group: closure, associativity, an additive identity 0, additive inverses, and commutativity. Next, we remove zero and look at F* = F \ {0}. The key upgrade from rings is that (F*,·) is also an abelian group: every nonzero element has a multiplicative inverse and there is a multiplicative identity 1. Then we connect the two operations with left and right distributivity. From there we hit core field facts like no zero divisors, and we list the standard examples: Q, R, C, and the finite fields F_p = Z/pZ when p is prime.
Then we move to subfields and the subfield test, show classic chains like Q ⊆ R ⊆ C, and talk about characteristic char(F). We define characteristic via repeated addition of 1, compute examples like char(Z/5Z)=5 and char(Q)=0, and explain the big theorem: the characteristic of a field is either a prime p or 0. Finally we preview field extensions, including Q(√2) and the finite field extension F4 built from F2[x]/(x^2 + x + 1), where α^2 = α + 1 forces every element to look like a + bα.

   • The Secret Structure Hidden Inside F2[x] m...  
   • An Infinite Group Example You Can Prove By...  
   • The Gateway to Group Theory: Groups in Und...  
   • The Backbone of Algebra: Rings In 40 Minut...  
   • The Hidden Glue: The Inverse Trick Powerin...  
   • Abstract Algebra  

PROPERTIES AND CONCEPTS USED
Field as a set with two operations + and ·
Abelian group axioms for (F,+)
Closure, associativity, identity 0, inverses, commutativity for addition
Nonzero set F* = F \ {0}
Abelian group axioms for (F*,·)
Multiplicative identity 1 and multiplicative inverses
Left and right distributive laws
Fields have no zero divisors
Examples Q, R, C
Finite fields F_p = Z/pZ for prime p
Subfield definition and inheritance of axioms
Subfield test using 1 ∈ K, a-b ∈ K, a·b^{-1} ∈ K
Characteristic char(R) via n·1 = 0
char(F) is prime p or 0
Subfields must share characteristic with the field
Prime subfield
Field extensions E/F
Adjoining √2 to get Q(√2)
Algebraic elements and solving x^2 - 2 = 0
Finite extension F4 from F2[x]/(x^2 + x + 1) and α^2 = α + 1
Normal form a + bα in F4

CHAPTERS
00:00 Introduction
00:50 Three Rockets
01:45 Field Definition
02:35 Addition Axioms
04:10 Multiplication On F*
06:40 Distributivity
07:55 Field Examples
09:25 Subfields
11:35 Subfield Test
13:05 Subfield Examples
14:25 Characteristic
16:55 Char Examples
18:40 Char Is Prime Or Zero
20:10 Proof Sketch
22:05 Char And Subfields
23:50 Prime Subfield
25:00 Field Extensions
26:30 Q(√2)
28:40 Solve x^2 - 2
30:45 Build F4
33:10 Elements Of F4
35:20 Wrap Up
36:19 Thanks For Watching

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