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 ${\displaystyle K}$ ${\displaystyle f\in K[X]}$

${\displaystyle L=K[X]/(f)}$

according to the ideal generated by a body. ${\displaystyle f}$

The polynomial has a zero, namely the image of . Therefore it is said, comes from by adjunction of a zero from and writes . ${\displaystyle f}$ ${\displaystyle L}$ ${\displaystyle \xi }$ ${\displaystyle X}$ ${\displaystyle L}$ ${\displaystyle K}$ ${\displaystyle \xi }$ ${\displaystyle f}$ ${\displaystyle K(\xi )=L}$

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: ${\displaystyle f}$ ${\displaystyle \mathbb {Q} ({\sqrt {2}})}$ ${\displaystyle f=X^{2}-2}$ ${\displaystyle f}$ ${\displaystyle 1}$ ${\displaystyle f}$ ${\displaystyle \mathbb {Q} ({\sqrt {2}})=\{a+b\cdot {\sqrt {2}}|a,b\in \mathbb {Q} \}}$

If the degree of is equal , then the elements of can be clearly identified in the form ${\displaystyle f}$ ${\displaystyle n}$ ${\displaystyle K(\xi )}$

${\displaystyle a_{n-1}\xi ^{n-1}+\ldots +a_{1}\xi +a_{0}}$with for i = 0.1 ..., n-1 ${\displaystyle a_{i}\in K}$

write.

The degree of body expansion is the same . ${\displaystyle [K(\xi ):K]}$ ${\displaystyle n}$

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 ${\displaystyle K}$ ${\displaystyle T}$ ${\displaystyle K(T)}$ ${\displaystyle K[T]}$

${\displaystyle {\frac {a_{n}T^{n}+\ldots +a_{1}T+a_{0}}{b_{m}T^{m}+\ldots +b_{1}T+b_{0}}}.}$

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 . ${\displaystyle R}$ ${\displaystyle R[X]/(f)}$ ${\displaystyle X}$ ${\displaystyle f\in R[X]}$ ${\displaystyle f}$

The transition from a ring to a polynomial ring is called the adjunction of an indeterminate. ${\displaystyle R}$ ${\displaystyle R[X]}$

Examples

• ${\displaystyle \mathbb {Z} \!\left[{\frac {1}{2}}\right]=\mathbb {Z} [X]/(2X-1)}$, the ring of rational numbers whose denominator is a power of two.
• ${\displaystyle \mathbb {Z} [{\sqrt {2}}]=\mathbb {Z} [X]/(X^{2}-2)}$, the ring of elements of that form ${\displaystyle \mathbb {Q} ({\sqrt {2}})}$

${\displaystyle \{a+b{\sqrt {2}}\mid a,b\in \mathbb {Z} \}}$

to have.

• ${\displaystyle \mathbb {Z} [X]/(X^{n}-1)}$; Ring homomorphisms from this ring into a ring correspond to the -th roots of unity in . ${\displaystyle R}$ ${\displaystyle n}$ ${\displaystyle R}$

See also

• Adjunction (single element)