Proof that K[x]/(f) is a field iff f is irreducible

Builds on Field Quotient ring Irreducible polynomial Irreducible element K[x] is a Euclidean domain Euclidean domains are principal ideal domains In a PID a nonzero element is irreducible iff it generates a maximal ideal R/M is a field iff M is maximal Maximal ideal Degree of a product Unit of a ring

Proof

Let $K$ be a field and $f \in K[x]$ .
The ring $K[x]$ is a Euclidean domain. So it is a principal ideal domain.
Case $f = 0$ . Then $(f) = \set{0}$ , and the quotient ring $K[x]/(f)$ is isomorphic to $K[x]$ through $g + (0) \mapsto g$ .
In $K[x]$ the element $x$ is not a unit: $x v = 1$ would give $1 + \deg v = 0$ by the degree of a product. So $K[x]/(0)$ is not a field.
Also $0$ is not irreducible, since an irreducible polynomial is nonzero by definition. So both sides of the equivalence are false, and the equivalence holds.
Case $f \ne 0$ . By definition, $f$ is irreducible over $K$ if and only if it is an irreducible element of $K[x]$ .
By Step 2 and the theorem on principal ideal domains, this holds if and only if $(f)$ is a maximal ideal of $K[x]$ .
Since $K[x]$ is commutative, [[ $K[x]/(f)$ is a field if and only if $(f)$ is maximal|0WD3SE7S1X]].
Steps 6 to 8 give: $K[x]/(f)$ is a field if and only if $f$ is irreducible over $K$ .

Idea

In a principal ideal domain, irreducible elements are exactly the nonzero generators of maximal ideals. A quotient by an ideal is a field exactly when the ideal is maximal. The zero polynomial is checked separately.

Reading path

To understand this page from the ground up, read these 75 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 R[x] is an integral domain iff R is, Proof of the degree of a product over an integral domain, Proof of strong induction from ordinary induction, Proof of the correspondence theorem for rings, Proof that R/M is a field iff M is maximal, Proof that the only ideals of a field are zero and the whole field, Proof of the well-ordering principle from strong induction, Proof that in a PID a nonzero element is irreducible iff it generates a maximal ideal, Proof that Euclidean domains are principal ideal domains, Proof that K[x] is a Euclidean domain.

Definition by recursion concept
Element definition
Group concept
Mathematical induction concept
Natural numbers object
Peano axioms concept
Proposition concept
Set definition
Ring definition
Successor function definition
Implication definition
Predicate definition
Ordered pair definition
Commutative ring definition
Polynomial ring definition
Unit of a ring definition
Zero divisor definition
Ideal definition
Addition of natural numbers definition
The successor function is injective statement
Zero is not a successor statement
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Degree of a polynomial definition
Field definition
Integral domain definition
Ideal generated by a subset definition
Subset definition
Proof of the commutativity of addition by induction proof
Maximal ideal definition
Order on the natural numbers definition
Two-sided ideal definition
Uniqueness quantifier definition
Leading coefficient definition
Proof of cancellation in an integral domain proof
Binary relation definition
Euclidean domain definition
Irreducible element definition
Proof that the only ideals of a field are zero and the whole field proof
Principal ideal definition
Commutativity of addition statement
Quotient ring definition
Proof of strong induction from ordinary induction proof
Monic polynomial definition
Proof of the degree of a product over an integral domain proof
Cancellation in an integral domain statement
Irreducible polynomial definition
The only ideals of a field are zero and the whole field statement
Principal ideal domain definition
Proof that R/M is a field iff M is maximal proof
Function definition
Strong induction statement
Division algorithm for polynomials statement
Degree of a product statement
Proof that in a PID a nonzero element is irreducible iff it generates a maximal ideal proof
Proof of the well-ordering principle from strong induction proof
Image of a set definition
Preimage of a set definition
Codomain definition
Injective function definition
Proof that R[x] is an integral domain iff R is proof
Well-ordering principle statement
In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
Surjective function definition
R[x] is an integral domain iff R is statement
Proof that Euclidean domains are principal ideal domains proof
Bijective function definition
Proof that K[x] is a Euclidean domain proof
Euclidean domains are principal ideal domains statement
Proof of the correspondence theorem for rings proof
K[x] is a Euclidean domain statement
Correspondence theorem for rings statement
R/M is a field iff M is maximal statement
Proof that K[x]/(f) is a field iff f is irreducible proof

Where this leads

53 pages build on this one.

Tag	`T9HJSB4ZAM`
URI	`m2://T9HJSB4ZAM`
Kind	proof
Writing	W0 stub
MSC	12E05
Proves	K[x]/(f) is a field iff f is irreducible
Uses	Field, Quotient ring, Irreducible polynomial, Irreducible element, K[x] is a Euclidean domain, Euclidean domains are principal ideal domains, In a PID a nonzero element is irreducible iff it generates a maximal ideal, R/M is a field iff M is maximal, Maximal ideal, Degree of a product, Unit of a ring
Methods	Proof by cases
Changed	4 November 2026

Proof that K[x]/(f) is a field iff f is irreducible

Proof

Idea

Reading path

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