Field theory and polynomials Galois theory Example

Quintic equation

Builds on Eisenstein's criterion Rolle's theorem Intermediate value theorem Solvable by radicals The symmetric group on five or more letters is not solvable Galois' theorem on solvability

Example

A quintic equation is an equation $ax^5 + bx^4 + cx^3 + dx^2 + ex + g = 0$ with $a \ne 0$ . Consider the quintic $f(x) = x^5 - 4x + 2$ over $\Q$ .

$f$ is irreducible over $\Q$ by Eisenstein's criterion at $2$ , as $2$ divides $-4$ and $2$ but not $1$ , and $4$ does not divide $2$ .
$f(-2) = -22$ , $f(0) = 2$ , $f(1) = -1$ and $f(2) = 26$ , so by the intermediate value theorem $f$ has at least three real roots.
$f'(x) = 5x^4 - 4$ has exactly two real zeros, so by Rolle's theorem $f$ has at most three real roots.
Hence $f$ has exactly three real roots and two nonreal roots.

By the theorem on prime degree polynomials with two nonreal roots, the Galois group of $f$ over $\Q$ is $S_5$ . Since $S_5$ is not solvable, Galois' theorem on solvability shows that $f$ is not solvable by radicals. By contrast, $x^5 - 2$ is solvable by radicals: its roots are $\sqrt[5]{2}\, \zeta^k$ , with $\zeta = e^{2\pi i/5}$ and $k = 0, \dots, 4$ .

Reading path

To understand this page from the ground up, read these 97 pages in order. The foundations come first.

Binary operation definition
Characteristic of a field concept
Cycle notation concept
Group concept
Limit point of a set definition
Local maximum definition
Radical extension concept
Sequence definition
Solvable by radicals concept
Commutativity definition
Commutator of two elements definition
Conjugate elements definition
Group homomorphism definition
Ring definition
Subgroup definition
Symmetric group definition
Limit of a function definition
Local minimum definition
Real sequence definition
Subsequence definition
Abelian group definition
Kernel of a homomorphism definition
Conjugate subgroup definition
Commutative ring definition
Disjoint cycles commute statement
Polynomial ring definition
Ring homomorphism definition
Subring definition
Transposition definition
Unit of a ring definition
Zero divisor definition
Left coset definition
Right coset definition
Subgroup criterion statement
Continuity at a point definition
Derivative definition
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Divisibility in a commutative ring definition
Normal subgroup definition
Decomposition of a permutation into disjoint cycles statement
Polynomial evaluation definition
Field definition
Ring isomorphism definition
Integral domain definition
Intersection of subgroups is a subgroup statement
Fermat's theorem on stationary points statement
Continuous function definition
Every sequence has a monotone subsequence statement
Convergent sequence definition
Monotone convergence theorem for sequences statement
Root of a polynomial definition
Associate elements definition
Prime element definition
Decomposition of a permutation into transpositions statement
Field of fractions definition
Subfield definition
Subgroup generated by a subset definition
Irreducible element definition
Normality via left and right cosets statement
Intermediate value theorem statement
Bolzano–Weierstrass theorem statement
Greatest common divisor in an integral domain definition
Split polynomial definition
Sign of a permutation definition
Commutator subgroup definition
Field extension definition
Unique factorisation domain definition
Irreducible polynomial definition
Quotient group definition
Boundedness theorem statement
Even and odd permutations definition
Content of a polynomial definition
Sign is a homomorphism statement
Sign of a cycle statement
Derived series definition
Galois group definition
Subnormal series definition
Intermediate field definition
Extreme value theorem statement
Alternating group definition
Primitive polynomial definition
Solvable group definition
Field generated by adjoining elements definition
Rolle's theorem statement
Product of primitive polynomials is primitive statement
Splitting field definition
The alternating group is generated by three-cycles statement
Subgroups of solvable groups are solvable statement
Simplicity of the alternating group statement
Galois group of a polynomial definition
Gauss's lemma statement
The symmetric group on five or more letters is not solvable statement
Eisenstein's criterion statement
Galois' theorem on solvability statement
Quintic equation example

Tag	`S1WS18QEGJ`
URI	`m2://S1WS18QEGJ`
Kind	example
Writing	W0 stub
MSC	12E12, 12F10
Illustrates	Abel–Ruffini theorem, Galois group of a prime degree polynomial with two nonreal roots
Uses	Eisenstein's criterion, Rolle's theorem, Intermediate value theorem, Solvable by radicals, The symmetric group on five or more letters is not solvable, Galois' theorem on solvability
Changed	2 November 2026

Quintic equation

Example

Reading path

Further reading

What this page needs for W1

What links here