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 . ${\displaystyle X}$ ${\displaystyle k}$ ${\displaystyle X}$ ${\displaystyle k}$ ${\displaystyle X/k}$ ${\displaystyle x\in X}$ ${\displaystyle \kappa (x)}$ ${\displaystyle k}$