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Squarefree integers determine quadratic fields.
Associated with any quadratic field is its ring of
integers (maximal order) and an equation order, and for every positive integer f
there exists an order of conductor f inside the maximal order.
For information on creating elements see
Section Creation of Elements.
Given an integer m that is not a square, create the field
Q(Sqrt(d)), where d is the squarefree part of m.
It is possible to assign a name to Sqrt(d) using
angle brackets: R<s> := QuadraticField(m).
Creation of the order Z[Sqrt(d)] in the quadratic
field F=Q(Sqrt(d)), with d squarefree.
IntegerRing(F) : FldQuad -> RngQuad
RingOfIntegers(F) : FldQuad -> RngQuad
Given a quadratic field F=Q(Sqrt(d)), with d squarefree, create
its maximal order. This order is Z[Sqrt(d)] if d ≡ 2, 3 mod 4
and Z[(1 + Sqrt(d)/2)] if d ≡ 1bmod4.
Given a quadratic order, this returns the quadratic field of which it
is an order.
Create the sub-order of index f in the order O of
a quadratic field. If O is maximal, this will be the unique
order of conductor f.
IsQuadratic(O) : RngOrd -> BoolElt, RngQuad
Return true if the field K or order O can be created
as a quadratic field or order and the quadratic field or order if so.
We create the quadratic field Q(Sqrt(5)) and an order in it, and
display some elements of the order in their representation as order
element and as field element.
> Q<z> := QuadraticField(5);
> Q eq QuadraticField(45);
true
> O<w> := sub< MaximalOrder(Q) | 7 >;
> O;
Order of conductor 7 in Q
> w;
w
> Q ! w;
1/2*(7*z + 7)
> Eltseq(w), Eltseq(Q ! w);
[ 0, 1 ]
[ 7/2, 7/2 ]
> ( (7/2)+(7/2)*z )^2;
1/2*(49*z + 147)
> Q ! w^2;
1/2*(49*z + 147)
> w^2;
7*w + 49
We define an injection φ:Q(Sqrt(5)) -> Q(ζ 5).
First a square root of 5 is identified in Q(ζ 5).
> Q<w> := QuadraticField(5);
> F<z> := CyclotomicField(5);
> C<c> := PolynomialRing(F);
> Factorization(c^2-5);
[
<c - 2*z^3 - 2*z^2 - 1, 1>,
<c + 2*z^3 + 2*z^2 + 1, 1>
]
> h := hom< Q -> F | -2*z^3 - 2*z^2 - 1 >;
> h(w)^2;
5
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