Existence of finite fields of every prime power order

Builds on Existence of splitting fields Formal derivative Repeated roots and the derivative Finite field

Statement

Let $p$ be a prime number and let $n \ge 1$ . Then there is a finite field of order $q = p^n$ .

Remarks

The proof: let $L$ be a splitting field of $f = x^q - x$ over $\mathbb{F}_p$ . The derivative $f' = q x^{q-1} - 1 = -1$ has no common root with $f$ , so $f$ has $q$ distinct roots in $L$ , by the criterion for repeated roots. The roots of $f$ form a subfield of $L$ , because $(a + b)^q = a^q + b^q$ and $(a b)^q = a^q b^q$ . This subfield has $q$ elements. So for each $n$ there is an irreducible polynomial of degree $n$ over $\mathbb{F}_p$ .

History

E. H. Moore first proved the classification of finite fields in 1893: there is a field of each prime power order, and all fields of a given order are isomorphic.

Reading path

To understand this page from the ground up, read these 60 pages in order. The foundations come first.

Group concept
Ring definition
Commutative ring definition
Ideal definition
Polynomial ring definition
Subring definition
Unit of a ring definition
Zero divisor definition
Divisibility in a commutative ring definition
Ideal generated by a subset definition
Maximal ideal definition
Two-sided ideal definition
Degree of a polynomial definition
Field definition
Formal derivative definition
Polynomial evaluation definition
Integral domain definition
Principal ideal definition
Quotient ring definition
Associate elements definition
Prime element definition
The only ideals of a field are zero and the whole field statement
Leading coefficient definition
Subfield definition
Finite field definition
Irreducible element definition
Field of fractions definition
Root of a polynomial definition
Euclidean domain definition
Correspondence theorem for rings statement
Greatest common divisor in an integral domain definition
Principal ideal domain definition
Irreducible polynomial definition
Monic polynomial definition
Field extension definition
Degree of a product statement
Split polynomial definition
Unique factorisation domain definition
Multiplicity of a root definition
R/M is a field iff M is maximal statement
In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
Division algorithm for polynomials statement
Content of a polynomial definition
Degree of a field extension definition
R[x] is an integral domain iff R is statement
Intermediate field definition
Repeated roots and the derivative statement
Primitive polynomial definition
Tower law statement
K[x] is a Euclidean domain statement
Field generated by adjoining elements definition
K[x]/(f) is a field iff f is irreducible statement
Product of primitive polynomials is primitive statement
Splitting field definition
Adjoining a root of a polynomial definition
Gauss's lemma statement
Polynomial ring over a UFD is a UFD statement
Every polynomial has a root in some extension field statement
Existence of splitting fields statement
Existence of finite fields of every prime power order statement

Where this leads

3 pages build on this one.

Tag	`0T8M3YC5RG`
URI	`m2://0T8M3YC5RG`
Kind	statement, theorem
Writing	W0 stub
MSC	12E20
Status	This copy of the site has no build data. Run `m2 status 0T8M3YC5RG`.
Uses	Existence of splitting fields, Formal derivative, Repeated roots and the derivative, Finite field

Existence of finite fields of every prime power order

Statement

Remarks

History

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

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