# Base field of Residue field

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.

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You forgot to pass map=True to get the maps.

K.<a> = CyclotomicField(11)
Q = K.primes_above(3)[0]
KQ = K.residue_field(Q)
VQ, fr, to = KQ.vector_space(map=True)


Then:

sage: to(a)
(0, 1, 0, 0, 0)

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