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I'm using Magma's online calculator to study some algebraic fuction fields, places, etc. I know that for an algebraic function field $F/K$, the places $\mathbb{P}_F$ are principal ideals of the valuation rings. But the generator is not unique. If, for a given $P\in\mathbb{P}_F$, $P=(t)$, then $P=(tu), \;\forall \, u\in\mathcal{O}_P^{\times}$, where $\mathcal{O}_P^{\times}$ is the group of units of the valuation ring $\mathcal{O}_P$.

So I have the following commands, wich I used in Magma's online calculator:

S:=GF(2);
R<x>:=FunctionField(S);
P<T>:=PolynomialRing(R);
h:=T^4-T-x;
F<a>:=FunctionField(h);
Places(F,1);

and the output after you press the submit button is

[ (1/x, 1/x*a^3), (x, a), (x, a + 1) ]

where each term in parentheses counts for one place of degree one. But I can't figure out what does it mean each of this terms, for example

(x, a)

What does each one of this comma separated values represent? I think that this has to do with the non-uniquiness of the generator, but I can't find the explanation on magma's documentation.

Magma's online calculator is freely availible at http://magma.maths.usyd.edu.au/calc/

Any help is appreciated.

Martin Sleziak
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Larara
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2 Answers2

3

The two elements are the two generators of the place as an ideal of a maximal order of the field. Magma uses orders and does not have valuation rings implemented.

See the documentation for TwoGenerators.

user187856
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  • I think I got it. For completeness, the documentation of TwoGenerators is: Return two elements of the function field of P which determine the place P. The sequence containing these two elements can be used as input to Place to create a place equal to P. – Larara Oct 28 '14 at 16:55
2

Just to let you know, I sent an email to the Magma group. The answer follows (it basically says what is in the answer of user187856).

You mean to say places are not principal ideals (in general)

A place corresponds to a maximal ideal in a maximal order of the function field. In Magma this is either MaximalOrderFinite(F) or MaximalOrderInfinite(F). The printing refers to generators of the ideal.

To work with them, use commands like (add these to the commands given in the question):

> Pl := Places(F,1);
>  Ideal(Pl[1]);
Prime Ideal of Maximal Order of F over Valuation ring of Univariate
rational function field over GF(2) with generator 1/x
Generators:
1/x
1/x*a^3
> Order(Ideal(Pl[1]));
Maximal Order of F over Valuation ring of Univariate rational function
field over GF(2) with generator 1/x
> Generators(Ideal(Pl[1]));
[
 1/x,
 1/x*a^3
]

Commands starting with > are input, and not starting with > are output.

Larara
  • 1,490