Field theory and polynomials Galois theory Theorem

Galois group of a prime degree polynomial with two nonreal roots

Builds on Irreducible polynomial Galois group of a polynomial Symmetric group Cauchy's theorem Transposition Complex conjugate Prime number

Statement

Let $p$ be a prime number and let $f \in \Q[x]$ be an irreducible polynomial of degree $p$ . Assume that $f$ has exactly two nonreal roots in $\C$ . Then the Galois group of $f$ over $\Q$ , acting on the $p$ roots of $f$ , is the full symmetric group $S_p$ .

Remarks

Let $G$ be the Galois group, seen as a group of permutations of the $p$ roots. Since $f$ is irreducible, adjoining one root gives a field of degree $p$ , so $p$ divides $\abs{G}$ . By Cauchy's theorem, $G$ contains an element of order $p$ , which is a $p$ -cycle. Complex conjugation restricts to an element of $G$ that swaps the two nonreal roots and fixes the others, so $G$ contains a transposition. For $p$ prime, a $p$ -cycle and any transposition generate $S_p$ . Example: $x^5 - 4x + 2$ has Galois group $S_5$ ; see the quintic equation. The prime degree matters: in degree $4$ , the group generated by $(1\,2\,3\,4)$ and $(1\,3)$ is a dihedral group of order $8$ , not $S_4$ .

Going deeper

For a student

Combined with Galois' theorem on solvability and the fact that $S_p$ is not solvable for $p \ge 5$ , the theorem proves the second part of the Abel–Ruffini theorem. The number of real roots can be found with calculus or with Sturm's theorem. The proof uses only that $\Q \subseteq \R$ ; it works for any subfield of $\R$ in place of $\Q$ .

Reading path

To understand this page from the ground up, read these 90 pages in order. The foundations come first. The path takes the proof Proof of Lagrange's theorem.

Complex numbers object
Cycle notation concept
Divisibility definition
Element definition
Group concept
Left coset concept
Proposition concept
Real and imaginary parts definition
Prime number definition
Set definition
Conjugate elements definition
Group homomorphism definition
Laws of exponents in a group statement
Order of a group definition
Ring definition
Subgroup definition
Symmetric group definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Complex conjugate definition
Empty set object
Family of sets definition
Ordered pair definition
Order of an element definition
Intersection definition
Transposition definition
Union definition
Conjugate subgroup definition
Group action definition
Commutative ring definition
Polynomial ring definition
Ring homomorphism definition
Subring definition
Unit of a ring definition
Zero divisor definition
Set-builder notation concept
Left coset definition
Right coset definition
Subgroup criterion statement
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Disjoint sets definition
P-group definition
Normal subgroup definition
Stabiliser definition
Orbit definition
Subset definition
Union of a family of sets definition
Polynomial evaluation definition
Field definition
Ring isomorphism definition
Integral domain definition
Intersection of subgroups is a subgroup statement
Index of a subgroup definition
Cosets have the same cardinality as the subgroup statement
Binary relation definition
Partition of a set definition
Fixed points of an action definition
Orbits partition the set statement
Transitive action definition
Normal core definition
Root of a polynomial definition
Order of a group is the index times the order of the subgroup statement
Subfield definition
Irreducible element definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Field extension definition
Split polynomial definition
Action on left cosets example
Irreducible polynomial definition
Equivalence relation definition
Intermediate field definition
Galois group definition
Orbit-stabiliser theorem statement
Field generated by adjoining elements definition
Equivalence class definition
Equivalence classes form a partition statement
Splitting field definition
Galois group of a polynomial definition
Proof of Lagrange's theorem proof
Lagrange's theorem statement
Size of an orbit times the size of the stabiliser statement
Fixed points of a p-group action statement
Cauchy's theorem statement
Galois group of a prime degree polynomial with two nonreal roots statement

Tag	`4CMPPTSJNP`
URI	`m2://4CMPPTSJNP`
Kind	statement, theorem
Writing	W0 stub
MSC	12F10, 12D10
Status	This copy of the site has no build data. Run `m2 status 4CMPPTSJNP`.
About	Galois group of a polynomial
Uses	Irreducible polynomial, Galois group of a polynomial, Symmetric group, Cauchy's theorem, Transposition, Complex conjugate, Prime number
Changed	2 November 2026

Galois group of a prime degree polynomial with two nonreal roots

Statement

Remarks

Going deeper

For a student

Reading path

Further reading

What this page needs for W1

What links here

Examples