Category:Field Theory

F

• [×] Field Automorphisms‎ (1 P)
• [×] Field Extensions‎ (16 P)
• [+] Field Homomorphisms‎ (2 C, 4 P)
• [+] Field Isomorphisms‎ (1 C, 1 P)

F cont.

• [×] Finite Fields‎ (9 P)

P

• [+] Polynomial Theory‎ (1 C, 56 P)
• [×] Prime Fields‎ (3 P)

Q

• [+] Quotient Fields‎ (1 C, 9 P)

R

• [+] Rational Numbers‎ (1 C, 38 P)

S

• [×] Subfields‎ (19 P)

T

• [×] Totally Ordered Fields‎ (3 P)

V

• [×] Valued Fields‎ (1 P)

C

• Characterisation of Non-Archimedean Absolute Values
• Characterisation of Totally Ordered Fields
• Characteristic of Field
• Characteristic of Field by Annihilator
• Commutative Division Ring is a Field

D

• Definition:Degree of Field Extension
• Division Theorem for Polynomial Forms over a Field

E

• Equivalent Absolute Values

F

• Field adjoined transcendental
• Field has No Proper Zero Divisors
• Field is Integral Domain
• Field is Subfield of Itself
• Field of Characteristic Zero has Unique Prime Subfield
• Field of Characteristic Zero has Unique Prime Subfield/Proof 1
• Field of Characteristic Zero has Unique Prime Subfield/Proof 2

F cont.

• Field of Complex Numbers
• Field of Rational Functions
• Field of Rational Numbers
• Field of Real Numbers
• Field Product Abelian Group
• Fields of Uncountable Cardinality K have Transcendence Degree K
• Finite Multiplicative Subgroup of Field is Cyclic

G

• Group of Units of Field

H

• Homomorphism from Field Either Monomorphism or Zero Homomorphism

I

• Ideals of a Field
• Integers Do Not Form a Field
• Integral Multiple Distributes over Ring Addition
• Integral Multiple of Integral Multiple
• Inverse of Multiplicative Inverse

L

• Lefschetz Principle (First-Order)

M

• Multiplicative Identity is Unique
• Multiplicative Inverse is Unique

N

• Numbers of Type Integer a plus b root 2 are Not a Field

P

• P-adic Valuation is Valuation
• P-adic Valuation is Well Defined
• P-adic Valuation on Integers
• Polynomial Forms over a Field is Principal Ideal Domain
• Polynomial Forms over Field is Euclidean Domain
• Power Function is Completely Multiplicative
• Primitive Roots of Unity
• Product of Integral Multiples

R

• Ring of Integers Modulo Prime is a Field

S

• Simplest Field