K(a) is isomorphic to K[x] modulo the minimal polynomial

Builds on Minimal polynomial of an algebraic element The minimal polynomial divides every polynomial vanishing at the element Evaluation homomorphism First isomorphism theorem for rings K[x]/(f) is a field iff f is irreducible Simple extension

Statement

Let $L/K$ be a field extension and let $a \in L$ be algebraic over $K$ with minimal polynomial $m_a$ . Then the evaluation map $K[x] \to L$ , $f \mapsto f(a)$ , induces a ring isomorphism $K[x]/(m_a) \cong K(a).$ In particular, $K(a) = K[a] = \{ f(a) : f \in K[x] \}$ .

Remarks

The proof: the kernel of the evaluation map is $(m_a)$ , by the divisibility property. The first isomorphism theorem gives $K[x]/(m_a) \cong K[a]$ . Since $m_a$ is irreducible, $K[x]/(m_a)$ is a field. So $K[a]$ is a field, and $K[a] = K(a)$ . If $a$ is transcendental over $K$ , then the evaluation map is injective, and it extends to an isomorphism $K(x) \cong K(a)$ .

Reading path

To understand this page from the ground up, read these 49 pages in order. The foundations come first.

Group concept
Ring definition
Commutative ring definition
Ideal definition
Polynomial ring definition
Ring homomorphism definition
Subring definition
Unit of a ring definition
Zero divisor definition
Divisibility in a commutative ring definition
Ideal generated by a subset definition
Maximal ideal definition
Two-sided ideal definition
Degree of a polynomial definition
Polynomial evaluation definition
Field definition
Integral domain definition
Kernel of a ring homomorphism definition
Ring isomorphism definition
Principal ideal definition
Quotient ring definition
The only ideals of a field are zero and the whole field statement
Evaluation homomorphism definition
Leading coefficient definition
Root of a polynomial definition
Subfield definition
Irreducible element definition
Euclidean domain definition
Correspondence theorem for rings statement
Principal ideal domain definition
Monic polynomial definition
Universal property of the quotient ring statement
Irreducible polynomial definition
Degree of a product statement
Field extension definition
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
Algebraic element definition
R[x] is an integral domain iff R is statement
First isomorphism theorem for rings statement
Intermediate field definition
Minimal polynomial of an algebraic element definition
K[x] is a Euclidean domain statement
Field generated by adjoining elements definition
The minimal polynomial divides every polynomial vanishing at the element statement
K[x]/(f) is a field iff f is irreducible statement
Simple extension definition
K(a) is isomorphic to K[x] modulo the minimal polynomial statement

Where this leads

16 pages build on this one.

Tag	`0ZN39QDTP9`
URI	`m2://0ZN39QDTP9`
Kind	statement, theorem
Writing	W0 stub
MSC	12F05
Status	This copy of the site has no build data. Run `m2 status 0ZN39QDTP9`.
Uses	Minimal polynomial of an algebraic element, The minimal polynomial divides every polynomial vanishing at the element, Evaluation homomorphism, First isomorphism theorem for rings, K[x]/(f) is a field iff f is irreducible, Simple extension

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