Adjunction (field theory)

Under adjunction is understood in the mathematical field of algebra adding more elements to a body or ring. In bodies we speak specifically of the Körperadjunktion and rings accordingly by the Ringadjunktion.

Adjunction of algebraic elements to a body

It was a body and an irreducible polynomial. Then the factor ring

To the ideal generated by a body.

The polynomial has a root, namely the image of. It is said therefore: results from by adjoining a root of and writes.

Often only implicitly included in the notation, for example, is meant in the polynomial. Normalizing the leading coefficient of, so is uniquely determined by the condition of irreducibility. It is found in this case an explicit representation of the body:

If the degree of the same, so the elements can be of uniquely in the form

. Write

The degree of the field extension is the same.

Adjunction transcendent elements to a body

If you want to expand a body around an element that should not be algebraically, it is called the adjunction of an indeterminate or a transcendent element. The resulting body is defined as the quotient of the polynomial ring body. Its elements are formal rational functions

Ringadjunktion

If instead of a general body before a commutative unitary ring, so it is also called extension by adjoining. The extensions are of the form with one variable and a polynomial. The behavior of such an extension does not depend critically on whether the leading coefficient of a unit of the ring, or, see entire element.

In the transition from one ring to the ring of polynomials is called the adjunction of indeterminates.

Examples

• , The ring of rational numbers whose denominator is a power of two.
• , The ring of elements of which form the

• ; Ring homomorphisms of this ring in a ring corresponding to the roots of unity in.