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

Builds on In a PID a nonzero element is irreducible iff it generates a maximal ideal R/M is a field iff M is maximal K[x] is a Euclidean domain Irreducible polynomial

Statement

Let $K$ be a field and let $f \in K[x]$ . Then the quotient ring $K[x]/(f)$ is a field if and only if $f$ is irreducible over $K$ .

Remarks

The proof: $K[x]$ is a principal ideal domain. So $(f)$ is maximal if and only if $f$ is irreducible, by the theorem on PIDs. Then $R/M$ is a field iff $M$ is maximal. For example, $\mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}$ and $\mathbb{Q}[x]/(x^2 - 2) \cong \mathbb{Q}(\sqrt{2})$ . The field $\mathbb{F}_2[x]/(x^2 + x + 1)$ has $4$ elements. If $f$ is irreducible of degree $n$ , then $K[x]/(f)$ is an extension of $K$ of degree $n$ . The class of $x$ in $K[x]/(f)$ is a root of $f$ .

Reading path

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Group concept
Ring definition
Commutative ring definition
Ideal definition
Polynomial ring definition
Unit of a ring definition
Zero divisor definition
Ideal generated by a subset definition
Maximal ideal definition
Two-sided ideal definition
Field definition
Degree of a polynomial definition
Integral domain definition
Principal ideal definition
Quotient ring definition
The only ideals of a field are zero and the whole field statement
Irreducible element definition
Leading coefficient definition
Euclidean domain definition
Correspondence theorem for rings statement
Irreducible polynomial definition
Principal ideal domain definition
Monic polynomial definition
Degree of a product statement
R/M is a field iff M is maximal statement
In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
Division algorithm for polynomials statement
R[x] is an integral domain iff R is statement
K[x] is a Euclidean domain statement
K[x]/(f) is a field iff f is irreducible statement

Where this leads

21 pages build on this one.

Tag	`0AWMPX35CW`
URI	`m2://0AWMPX35CW`
Kind	statement, theorem
Writing	W0 stub
MSC	12E05
Status	This copy of the site has no build data. Run `m2 status 0AWMPX35CW`.
Uses	In a PID a nonzero element is irreducible iff it generates a maximal ideal, R/M is a field iff M is maximal, K[x] is a Euclidean domain, Irreducible polynomial

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