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Field Extension
The main study of Field Theory
By: Valerie Toothman
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
Definitions
• Field - any set of elements that satisfies the
field axioms
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]
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
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
One Happy Family!
Field Extension
Algebraic Extension
Finite Extension
Galois Extension
(Normal and Separable extension)
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.
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
The main study of Field Theory
By: Valerie Toothman