# Category:Field Extensions

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This category contains results about **Field Extensions**.

Definitions specific to this category can be found in Definitions/Field Extensions.

Let $F$ be a field.

A **field extension over $F$** is a field $E$ where $F \subseteq E$.

That is, such that $F$ is a subfield of $E$.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

## Pages in category "Field Extensions"

The following 34 pages are in this category, out of 34 total.

### A

- Abstract Model of Algebraic Extensions
- Algebraic Closure of Field is Unique
- Algebraic Element of Degree 3 is not Element of Field Extension of Degree Power of 2
- Algebraically Closed Field is Perfect
- Automorphism Group Acts Faithfully on Generating Set
- Automorphism Group of Complex Numbers over Real Numbers

### D

- Decomposition of Field Extension as Separable Extension followed by Purely Inseparable
- Degree of Element of Finite Field Extension divides Degree of Extension
- Degree of Field Extensions is Multiplicative
- Degree of Simple Algebraic Field Extension equals Degree of Algebraic Number
- Doubling the Cube by Compass and Straightedge Construction is Impossible

### E

### F

- Field Adjoined Algebraic Elements is Algebraic
- Field Adjoined Set
- Field Adjoined Set/Corollary
- Field Norm of Complex Number Equals Field Norm
- Finite Field Extension is Algebraic
- Finite Orbit under Group of Automorphisms of Field implies Separable over Fixed Field
- Finite-Dimensional Integral Domain over Field is Field
- Finitely Generated Algebraic Extension is Finite