1. The main study of Field Theory By: Valerie Toothman
Field Extension
The main study of Field Theory
By: Valerie Toothman

2. What is a Field Extension?
Abstract Algebra
Main object of study in field theory
The General idea is to start with a field and construct a larger field that contains that original field and satisfies additional properties

3. Definitions
Field - any set of elements that satisfies the field axioms

4. Definitions
Subfield – Let L be a field and K be a subset of L. If the subset K of L is closed under the field operations inherited from L, then the subset K of L is a subfield of L.
Extension Field- If K is a subfield of L then the larger field L is said to be the extension field of K.
Notation – L/K (L over K) signifies that L is an extension field of K
Degree – The field L can be considered as a vector space over the field K. The dimension of this vector space is the degree denoted by [L:K]

5. Example
The field of complex numbers C is an extension field of the field of real numbers R, and R in turn is an extension field of the field of rational numbers Q.
C- a+bi where a is real a number
R – includes all rational numbers
So we say C/R , R/Q, and C/Q

6. Example
The set Q(√2) = {a + b√2 | a, b ∈ Q} is an extension field of Q.
Degree - √2 is a root of 𝑥 2 -2 which cannot be factored in Q[x] so we use {1, √2} as a basis. Therefore the degree is 2

7. One Happy Family!
Field Extension Algebraic Extension Finite Extension Galois Extension (Normal and Separable extension)

8. Galois Theory
Galois theory- the study of algebraic extensions of a field. Algebraic extensions is a kind of field extension (L/K) that for every element of L is a root of some non-zero polynomial with coefficients in K.
In General it provides a connection between field theory and group theory by Roots of a given polynomial.

9. The Theory of Field extensions (including Galois theory)
Leads to impossibility proofs of classical problems such as angle trisection and squaring the circle with a compass and straightedge

10. Field Extensions
The main study of Field Theory
By: Valerie Toothman