Let be a field.
Let be a fixed algebraic closure of .
The extension is geometric, but is not geometric.
Theorem 1 (Thm. I.6.9 of ).
Let be a function field over a field . Let which is a function field over , a fixed algebraic closure of , and let be the curve given by the previous theorem. The genus of is, by definition, the genus of .
Let be the rational function field over and let . The prime ideals of are generated by monic irreducible polynomials in . Let be such a prime. Then , the localization of at the prime is a discrete valuation ring with and the quotient field of is . Thus is a prime of .
One can define an ‘extra’ prime in the following way. Let and let be the prime ideal of generated by . The localization ring is a prime of , called the prime at infinity.
|Date of creation||2013-03-22 15:34:35|
|Last modified on||2013-03-22 15:34:35|
|Last modified by||alozano (2414)|
|Synonym||algebraic function field|
|Defines||rational function field|
|Defines||genus of a function field|
|Defines||degree of a prime|