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Math 2.0

Field theory and polynomials Proof

Proof that finite subgroups of the multiplicative group of a field are cyclic

Builds on Field Group of units of a ring Abelian group Exponent of a group g to the order of the group is the identity A polynomial over a field has at most as many roots as its degree A finite abelian group is cyclic iff its exponent equals its order

Proof

Let $K$ be a field, and let $G$ be a finite subgroup of the group of units $K^\times$ . Let $n = \abs{G}$ .
Multiplication in $K$ is commutative, so $G$ is an abelian group.
Every $g \in G$ satisfies $g^n = 1$ . So some positive integer $m$ has $g^m = 1$ for all $g \in G$ .
Let $e$ be the exponent of $G$ : the least positive integer with $g^e = 1$ for all $g \in G$ . By step 3, $e \le n$ .
Every element of $G$ is a root in $K$ of the polynomial $x^e - 1 \in K[x]$ , which is nonzero of degree $e$ .
This polynomial has at most $e$ roots in $K$ . The $n$ elements of $G$ are distinct roots, so $n \le e$ .
By steps 4 and 6, the exponent $e$ equals $\abs{G} = n$ .
A finite abelian group whose exponent equals its order is cyclic. So $G$ is cyclic.

Idea

In a field, the equation $x^e = 1$ has at most $e$ solutions. So the exponent of a finite subgroup of $K^\times$ cannot be smaller than its order. For an abelian group, exponent equal to order means that one element has the full order.

Reading path

To understand this page from the ground up, read these 136 pages in order. The foundations come first. The path takes the proofs Proof of Bézout's identity, Proof of cancellation in an integral domain, Proof of the generalized Euclid's lemma, 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 product of two finite cyclic groups is cyclic iff their orders are coprime, Proof that a polynomial over a field has at most as many roots as its degree, Proof of the remainder theorem, Proof that a finite abelian group is cyclic iff its exponent equals its order, 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 that cyclic groups of the same order are isomorphic, 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
Vector space definition
Commutativity definition
Identity element definition
Greatest common divisor definition
Set definition
Direct product of groups definition
Group homomorphism definition
Order of a group definition
Ring definition
Subgroup definition
Successor function definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
External direct sum definition
Inverse element definition
Abelian group definition
Coprime integers definition
Empty set object
Family of sets definition
Ordered pair definition
Group isomorphism definition
Intersection definition
Addition of natural numbers definition
Union definition
Commutative ring definition
Polynomial ring definition
Ring homomorphism definition
Unit of a ring definition
Zero divisor definition
Left coset definition
Right coset 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
Disjoint sets definition
Order on the natural numbers definition
Proof of the commutativity of addition by induction proof
Group of units of a ring definition
Divisibility in a commutative ring definition
Subset definition
Degree of a polynomial definition
Polynomial evaluation definition
Union of a family of sets definition
Field definition
Axiom of extensionality statement
Integral domain definition
Uniqueness quantifier definition
Uniqueness of inverses statement
Binary relation definition
Commutativity of addition statement
Partition of a set definition
Leading coefficient definition
Root of a polynomial definition
Proof of cancellation in an integral domain proof
Evaluation homomorphism definition
Proof of strong induction from ordinary induction proof
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Proof of the laws of exponents in a group proof
Monic polynomial definition
Strong induction statement
Multiplicity of a root definition
Function 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
Equivalence relation definition
Laws of exponents in a group statement
Proof of the well-ordering principle from strong induction proof
Codomain definition
Injective function definition
Division algorithm for polynomials statement
Degree of a product statement
Order of an element definition
Well-ordering principle statement
Cyclic subgroup generated by an element definition
Surjective function definition
Remainder theorem statement
Equivalence class definition
Proof that R[x] is an integral domain iff R is proof
Exponent of a group definition
Proof of the division algorithm proof
Cyclic group definition
Bijective function definition
Proof that equivalence classes form a partition proof
Proof of the factor theorem proof
R[x] is an integral domain iff R is statement
Division algorithm statement
Equinumerous sets definition
Equivalence classes form a partition statement
Factor theorem statement
Cardinality definition
Proof of Bézout's identity proof
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
Bézout's identity 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 of the generalized Euclid's lemma 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
Generalized Euclid's lemma statement
Order of a cyclic subgroup equals the order of its generator statement
Proof of Lagrange's theorem proof
Proof that a product of two finite cyclic groups is cyclic iff their orders are coprime proof
Proof that cyclic groups of the same order are isomorphic proof
Lagrange's theorem statement
Proof that the order of an element divides the order of the group proof
Cyclic groups of the same order are isomorphic statement
Product of cyclic groups is cyclic iff the orders are coprime statement
Order of an element divides the order of the group statement
Proof that g to the order of the group is the identity proof
Classification of finite abelian groups statement
g to the order of the group is the identity statement
Proof that a finite abelian group is cyclic iff its exponent equals its order proof
A finite abelian group is cyclic iff its exponent equals its order statement
Proof that finite subgroups of the multiplicative group of a field are cyclic proof

Where this leads

18 pages build on this one.

Further reading

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

About this page

Tag	`A2VZWSBS4B`
URI	`m2://A2VZWSBS4B`
Kind	proof
Writing	W0 stub
MSC	12E20
Proves	Finite subgroups of the multiplicative group of a field are cyclic
Uses	Field, Group of units of a ring, Abelian group, Exponent of a group, g to the order of the group is the identity, A polynomial over a field has at most as many roots as its degree, A finite abelian group is cyclic iff its exponent equals its order
Changed	4 November 2026

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