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### Keyword

Finite and Infinite Maximal Orders

### Description

According to their coefficient rings 'k[T]' or 'O at infinity' orders are called finite or infinite. By an equation order (or coordinate ring) over 'k[T]' we mean the quotient ring 'k[T][y] / f(T,y)k[T][y]'. Equation orders over 'O at infinity' are defined analogously for suitable, field generating polynomials (See examples for different orders).
One can define elements of orders by calling 'Element'. Since the orders have bases, it is enough to specify coefficients of linear combinations of the basis elements (see example). Afterwards one can perform the operations with these elements as usual.
Usually one wants to work with the maximal orders since only these are Dedekind rings. For convenience there is a function which expects the defining polynomial and which first checks for irreducibility and separability and defines then the algebraic function field 'F' and the maximal orders 'o' and 'oi' (see example below).

### Examples

ff := FiniteField(5); fx := FunctionField(ff); fxy := PolynomialAlgebra(fx); F := FunctionField(fxy.1^3+fx.1^4+1); o:=MaximalOrderFinite(F); oi:=MaximalOrderInfinite(F); a:=Element(o,[0,1,0]); b:=Element(oi,[0,1/fx.1,1/fx.1^2+1]); a^3+fx.1^4+1; a+b; Coerce(o,a);
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