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

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

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}$