Abhishek's profile - activity

Given a number field $L$, I can construct a vector space $V$ over the rational field as follows :

V,fr,to = L.vector_space()

Now, if I want to define the same vector space over a subfield of $L$, say $K$ rather than the rational field, is there any command.

Thanks a lot, didn't know of this function.

I have a list L, and would like to construct all polynomials of degree d, with coefficients from the list. An example is the following :

SET=[];
K.<a>=FiniteField(17);
F.<x> = PolynomialRing(K,'x');
for f in F.monics( of_degree=8):
    S=f.coefficients(sparse=False);
    if S==([K(1),K(16)]):
        SET=SET+[f];
print SET;

The above command is too time consuming. Is there a simpler way, when we can get the output in less time.

Let $K$ be a number field, $O_K$ be its ring of integers, $p$ prime ideal in $\mathbb{Z}$ and $\mathfrak{p}$ be a prime ideal above $p$. I am trying to construct the $\mathbb{Z}/p\mathbb{Z}$ vector space $O_K/\mathfrak{p}$, and a $\mathbb{Z}/p\mathbb{Z}$ subspace spanned by certain images of elements of $O_K$.

For constructing the prime, I am able to use Q=K.primes_above(p)[0], but I do not know how to view the residue field F=K.residue_field(Q) as the vector space over $\mathbb{Z}/p\mathbb{Z}$.

The command V,fr,to=F.vector_space() indicates $\textbf{ValueError: too many values to unpack}$. Here K= $\mathbb{Q}(\zeta_{11})$ and $p=3$. Can someone suggest me alternative commands for the same.

I would like to construct a system of linear equations as follows and find a basis of the solutions : Fix $p$, $q$ $$a_{i, j} - 3a_{i-1,j} - 4a_{i,j-1} +10a_{i-1,j-1} = 0$$ for all $$0 \le i \le \frac{p-1}{2} - 1 ; 0 \le j \le q-2$$ with further conditions : $ a_{-1,j} = -a_{\frac{p-1}{2}-1, j}, a_{i,-1} = a_{i,q-2} $ for all $i,j$ satisfying the above conditions

Further more, is there a way that the same can be implemented for $a_{i,j,k}$ and so on.

Consider a field L containing a subfield F. I would like to look at L as a F vector space without using the command relativise. Is there any way to obtain this.

For example : Let L.=CyclotomicField(53*52), and F.=CyclotomicField(53).

I have 2 isomorphic number fields L.and M., and I would like to define an embedding from L to M sending a to b. Is this possible without using the command L.embeddings(M)?