# Category:Field Theory

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

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

**Field Theory** is a branch of abstract algebra which studies fields and other related algebraic structures.

## Subcategories

This category has the following 40 subcategories, out of 40 total.

### C

- Cancellation Laws (14 P)

### D

### E

### F

- Field Automorphisms (1 P)
- Field is Integral Domain (3 P)

### G

### K

- Kummer Theory (empty)

### M

### O

- Ordered Fields (6 P)

### P

- Perfect Fields (2 P)
- Prime Fields (3 P)

### R

### S

- Separable Polynomials (empty)

### T

- Topological Fields (2 P)
- Totally Ordered Fields (3 P)

### V

- Valuation Rings (1 P)
- Valued Fields (1 P)

### Z

- Zero of Field is Unique (3 P)

## Pages in category "Field Theory"

The following 61 pages are in this category, out of 61 total.

### A

### C

- Cancellation Law for Field Product
- Characterisation of Ordered Fields
- Common Index Law for Field
- Commutative and Unitary Ring with 2 Ideals is Field
- Complete Archimedean Valued Field is Real or Complex Numbers
- Condition for Difference of Field Elements to be Zero
- Condition for Division by Field Elements to be Unity
- Construction of Direct Product of Fields

### F

- Field Contains at least 2 Elements
- Field has 2 Ideals
- Field has Algebraic Closure
- Field has no Proper Zero Divisors
- Field is Integral Domain
- Field is Principal Ideal Domain
- Field of Rational Functions is Field
- Field of Uncountable Cardinality K has Transcendence Degree K
- Field Product with Non-Zero Element yields Unique Solution
- Field Product with Zero
- Field Unity Divided by Element equals Multiplicative Inverse
- Finite Multiplicative Subgroup of Field is Cyclic
- Finite Ring with Multiplicative Norm is Field
- Finite Ring with No Proper Zero Divisors is Field
- Frobenius Endomorphism on Field is Injective

### I

### M

### N

### P

- Polynomial Forms over Field form Integral Domain/Formulation 1
- Polynomial Forms over Field form Principal Ideal Domain
- Polynomial Forms over Field is Euclidean Domain
- Power Function is Completely Multiplicative
- Powers of Field Elements Commute
- Product of Field Negatives
- Product of Indices Law for Field
- Product of Integral Multiples
- Product with Field Negative
- Product with Field Negative/Corollary