# Residual class field

Residual class fields play an important role in various areas of algebra and number theory . In their simplest form they are the mathematical abstraction of the remainder when dividing by a prime number , in algebraic geometry they appear when the local structure of a geometric object is described in a point.

## definition

Be a ring with a maximum ideal . Then is the quotient ring , which is a factor ring of a maximal ideal one body, the residual body of respect . ${\ displaystyle A}$ ${\ displaystyle {\ mathfrak {m}}}$ ${\ displaystyle A / {\ mathfrak {m}}}$${\ displaystyle A}$${\ displaystyle {\ mathfrak {m}}}$

## Examples

### Residual class field modulo a prime number

Be the ring of whole numbers . Since there is a main ideal ring , maximum ideals of just the ideals generated by prime elements . So if a prime number , the remainder class ring is a field , more precisely a finite field with elements. It is called the remainder class field modulo and is usually denoted by. Note, however, that there are also finite fields , which have nothing to do with the respective residue class rings. ${\ displaystyle A = \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle p}$ ${\ displaystyle \ mathbb {Z} / p \ mathbb {Z}}$${\ displaystyle p}$${\ displaystyle p}$${\ displaystyle \ mathbb {F} _ {p}}$${\ displaystyle \ mathbb {F} _ {p ^ {2}}}$${\ displaystyle \ mathbb {F} _ {p ^ {3}}, \ ldots}$

For more details on finite fields see finite field .

### Residual class field of local rings

Be a local ring , i.e. a ring in which there is only one maximum ideal . Then there is only one remainder class field, namely , and we speak of the remainder class field of . ${\ displaystyle A}$${\ displaystyle {\ mathfrak {m}}}$${\ displaystyle A}$${\ displaystyle A / {\ mathfrak {m}}}$${\ displaystyle A}$

#### Residual class field of discrete evaluation rings

Be the evaluation ring of a discretely evaluated body . Then there is a local main ideal ring such that the maximum ideal of is generated by an element . Such an element is called a uniformizing element and in this case it is also called a remainder class field of . ${\ displaystyle {\ mathcal {O}}}$ ${\ displaystyle K}$${\ displaystyle {\ mathcal {O}}}$${\ displaystyle {\ mathcal {O}}}$${\ displaystyle \ pi}$${\ displaystyle {\ mathcal {O}} / (\ pi)}$${\ displaystyle K}$

#### Residual class field of points on schemas

Be a scheme with a point . Then the remainder class field of the local ring is called the remainder class field of in and is usually denoted by. ${\ displaystyle X}$${\ displaystyle x \ in X}$${\ displaystyle {\ mathcal {O}} _ {X, x}}$${\ displaystyle X}$${\ displaystyle x}$${\ displaystyle \ kappa (x)}$

If a schema is over a field , then all remainder class fields are of field extensions of . Is locally finite type and is a closed point, then is a finite extension of . This is essentially the statement of Hilbert's zero theorem . ${\ displaystyle X}$${\ displaystyle k}$${\ displaystyle X}$${\ displaystyle k}$${\ displaystyle X / k}$ ${\ displaystyle x \ in X}$${\ displaystyle \ kappa (x)}$${\ displaystyle k}$