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Math 2.0

Field theory and polynomials Proof

Proof that every element of a finite field of order q satisfies x^q = x

Builds on Finite field Group of units of a ring g to the order of the group is the identity A polynomial over a field has at most as many roots as its degree Degree of a polynomial

Proof

Let $F$ be a finite field with $q$ elements. Let $F^\times$ be its group of units.
In a field, every nonzero element is a unit. So $F^\times = F \setminus \set{0}$ , and $F^\times$ is a group of order $q - 1$ .
Let $a \in F$ with $a \neq 0$ . Then $a^{q-1} = 1$ , since $a$ lies in the finite group $F^\times$ of order $q - 1$ .
Multiplying by $a$ gives $a^q = a$ .
For $a = 0$ we have $a^q = 0 = a$ , since $q \ge 1$ . So $a^q = a$ for every $a \in F$ .
Let $f = x^q - x$ and $g = \prod_{a \in F} (x - a)$ in $F[x]$ . Both are monic of degree $q$ .
So the leading terms cancel in $h = f - g$ , and $h$ is $0$ or has degree less than $q$ .
Every $a \in F$ is a root of $f$ by step 5. It is also a root of $g$ , because $g$ has the factor $x - a$ . So every $a \in F$ is a root of $h$ .
Thus $h$ has $q$ distinct roots in $F$ . A nonzero polynomial of degree less than $q$ has fewer than $q$ roots. So $h = 0$ .
Hence $x^q - x = \prod_{a \in F} (x - a)$ in $F[x]$ .

Idea

The nonzero elements form a group of order $q - 1$ , so Lagrange's theorem gives $a^{q-1} = 1$ for them. Then all $q$ elements are roots of $x^q - x$ . A monic polynomial of degree $q$ with $q$ known distinct roots is the product of the linear factors, because the difference would have too many roots.

Reading path

To understand this page from the ground up, read these 115 pages in order. The foundations come first. The path takes the proofs Proof of cancellation in an integral domain, Proof of the commutativity of addition by induction, Proof that equivalence classes form a partition, Proof that g to the n equals 1 iff the order divides n, Proof that R[x] is an integral domain iff R is, Proof that g to the order of the group is the identity, Proof that the order of an element divides the order of the group, Proof of the degree of a product over an integral domain, Proof of strong induction from ordinary induction, Proof of the laws of exponents in a group, Proof that a polynomial over a field has at most as many roots as its degree, Proof of the remainder theorem, Proof that the order of a cyclic subgroup equals the order of its generator, Proof of the well-ordering principle from strong induction, Proof of the division algorithm, Proof of the uniqueness of inverses, Proof of Lagrange's theorem, Proof of the factor theorem, Proof that cosets have the same cardinality as the subgroup.

Binary operation definition
Definition by recursion concept
Divisibility definition
Element definition
Group concept
Left coset concept
Mathematical induction concept
Natural numbers object
Peano axioms concept
Proposition concept
Identity element definition
Set definition
Order of a group definition
Ring definition
Subgroup definition
Successor function definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Inverse element definition
Empty set object
Family of sets definition
Ordered pair definition
Commutative ring definition
Polynomial ring definition
Ring homomorphism definition
Unit of a ring definition
Zero divisor definition
Intersection definition
Union definition
Left coset definition
Right coset definition
Addition of natural numbers definition
Set-builder notation concept
The successor function is injective statement
Zero is not a successor statement
Biconditional definition
Existential quantifier definition
Universal quantifier definition
Proof of the uniqueness of inverses proof
Cartesian product definition
Divisibility in a commutative ring definition
Degree of a polynomial definition
Disjoint sets definition
Polynomial evaluation definition
Field definition
Group of units of a ring definition
Integral domain definition
Subset definition
Proof of the commutativity of addition by induction proof
Axiom of extensionality statement
Order on the natural numbers definition
Union of a family of sets definition
Uniqueness quantifier definition
Uniqueness of inverses statement
Leading coefficient definition
Binary relation definition
Proof of cancellation in an integral domain proof
Root of a polynomial definition
Evaluation homomorphism definition
Finite field definition
Partition of a set definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Monic polynomial definition
Proof of the laws of exponents in a group proof
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Multiplicity of a root definition
Proof of the degree of a product over an integral domain proof
Proof of the remainder theorem proof
Cancellation in an integral domain statement
Function definition
Strong induction statement
Laws of exponents in a group statement
Division algorithm for polynomials statement
Equivalence relation definition
Degree of a product statement
Codomain definition
Injective function definition
Proof of the well-ordering principle from strong induction proof
Cyclic subgroup generated by an element definition
Order of an element definition
Remainder theorem statement
Proof that R[x] is an integral domain iff R is proof
Surjective function definition
Equivalence class definition
Well-ordering principle statement
Proof of the factor theorem proof
Bijective function definition
R[x] is an integral domain iff R is statement
Proof that equivalence classes form a partition proof
Proof of the division algorithm proof
Equinumerous sets definition
Factor theorem statement
Division algorithm statement
Equivalence classes form a partition statement
Cardinality definition
Proof that g to the n equals 1 iff the order divides n proof
Proof that a polynomial over a field has at most as many roots as its degree proof
g to the n equals 1 iff the order divides n statement
A polynomial over a field has at most as many roots as its degree statement
Proof that cosets have the same cardinality as the subgroup proof
Proof that the order of a cyclic subgroup equals the order of its generator proof
Cosets have the same cardinality as the subgroup statement
Order of a cyclic subgroup equals the order of its generator statement
Proof of Lagrange's theorem proof
Lagrange's theorem statement
Proof that the order of an element divides the order of the group proof
Order of an element divides the order of the group statement
Proof that g to the order of the group is the identity proof
g to the order of the group is the identity statement
Proof that every element of a finite field of order q satisfies x^q = x proof

Where this leads

15 pages build on this one.

Further reading

Web https://en.wikipedia.org/wiki/Finite_field, Multiplicative structure

About this page

Tag	`WF7HV1W1BH`
URI	`m2://WF7HV1W1BH`
Kind	proof
Writing	W0 stub
MSC	12E20
Proves	Every element of a finite field of order q satisfies x^q = x
Uses	Finite field, Group of units of a ring, g to the order of the group is the identity, A polynomial over a field has at most as many roots as its degree, Degree of a polynomial
Changed	4 November 2026

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