# Adjunction (algebra)

In the mathematical sub-area of algebra, **adjunction** is the addition of further elements to a body or ring . In the case of bodies, one speaks specifically of the **body adjunction** and in the case of rings accordingly of the **ring adjunction** .

## Adjunction of algebraic elements to form a body

Let it be a field and an irreducible polynomial . Then is the factor ring

according to the ideal generated by a body.

The polynomial has a zero, namely the image of . Therefore it is said, comes from by *adjunction* of a zero from and writes .

Often it is only implicit in the notation, for example the polynomial is meant. Normalizing the leading coefficient of on , so is the irreducible uniquely determined by the condition. In this case there is an explicit representation of the body:

If the degree of is equal , then the elements of can be clearly identified in the form

- with for i = 0.1 ..., n-1

write.

The degree of body expansion is the same .

## Adjunction of transcendent elements into one body

If you want to add an element to a body that is not supposed to be algebraic, one speaks of the adjunction of an indefinite or a transcendent element . The resulting body is defined as the quotient body of the polynomial ring . Its elements are formal rational functions

## Ring adjunct

If instead of a body there is a commutative, unitary ring , then one speaks of expansion through adjunction. The extensions are of the form with an indeterminate and a polynomial . The behavior of such an extension depends crucially on whether the guide coefficient of is a unit of the ring or not, *see *whole element .

The transition from a ring to a polynomial ring is called the adjunction of an indeterminate.

### Examples

- , the ring of rational numbers whose denominator is a power of two.
- , the ring of elements of that form

- to have.

- ; Ring homomorphisms from this ring into a ring correspond to the -th roots of unity in .