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Is was wondering if, given a field $F$ with a known basis and an element $b$ which is algebraic over that field, it is possible to construct explicitly a basis for $F[b]$, the extension of $F$ by $b$.

Suppose, for example, we're considering the field $\mathbf Z/3\mathbf Z$, for which $\{1\}$ is a basis, and $b$ is a root of the polynomial $x^3+x^2+x+1\in Z/3Z[x]$. Could we construct a basis for $\mathbf Z/3\mathbf Z[b]$, and, if so, how might I even begin to set it up?

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You have to find the minimal polynomial $p(x)$ of $b$ over $\mathbb{Z_{3}}$; then if $\deg(p(x) = n$ a basis for $\mathbb{Z_{3}}[b]$ over $\mathbb{Z_{3}}$ will be $$\lbrace 1, b, \ldots b^{n-1}\rbrace$$ In our case $x^3+x^2+x+1\in \mathbb{Z_{3}}[x]$ is the minimal polynomial of $b$ if and only if it is irreducible over $\mathbb{Z_{3}}$, i.e. if and only if has no roots in $\mathbb{Z_{3}}$, because its $\deg$ is $3$

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