Define an order in a number field

Usually, to create an order, one starts with a number field respectively a dfining polynomial:

julia> Qx, x = QQ[:x];

julia> K, a = number_field(x^2 - 5, :a);

The number field is $K=\text{Q}(\alpha )$, where $\alpha$ is a root of $f=x^2-10$. Since the polynomial $f$ is integral, the ring $\text{Z}[\alpha ]$ generated by $\alpha$ is an order of $K$. This is so-called equation order can be defined as follows:

julia> E = equation_order(K)
Order of number field of degree 2 over QQ
with Z-basis [1, a]

The maximal order can be computed as:

julia> maximal_order(K)
Maximal order of number field of degree 2 over QQ
with basis [1, 1//2*a + 1//2]

We can also compute the order generated by a finite set of elements, that is, the smallest order of $K$ containing all elements:

julia> order(K, [15*a, 10*a + 10])
Order of number field of degree 2 over QQ
with Z-basis [1, 5*a]

We can also supply a matrix containing the coordinates of a basis (as rows):

julia> order(K, ZZ[1 0; 0 5])
Order of number field of degree 2 over QQ
with Z-basis [1, 5*a]