Polynomial Long Division over GF(p)

I'm afraid there isn't such a package to do calculations over finite fields. Arithmetic over finite fields GF(p^n) may be too complex for TeX. GF(2) and GF(p) are much easier, but there seems no such a package either.

To typeset long division manually, you can simply use an array. For example:

\documentclass{article}
\usepackage{amssymb}

\begin{document}

\[
x^3 - x + 1 = (x-1)(x^2+x) + 1 \in \mathbb{F}_3[x]
\]

\[
\renewcommand\arraystretch{1.2}
\begin{array}{*2r @{\hskip\arraycolsep}c@{\hskip\arraycolsep} *4r}
  &    &&    &  1 &  1 & 0 \\
\cline{3-7}
1 & -1 &\big)& 1 &  0 & -1 & 1 \\
  &    &&  1 & -1 \\
\cline{4-6}
  &    &&    &  1 & -1 \\
  &    &&    &  1 & -1 \\
\cline{5-7}
  &    &&    &    &    & 1 \\
\end{array}
\]

\end{document}

If you need a lot of these long divisions, it is worth to write a program (not necessarily in TeX) to generate the code. It's not very difficult for polynomials over GF(p).

---

For edited question:

To increase the vertical space between array lines, you can redefine the factor \arraystretch as showed above. Or you can use makecell package to add a gap.

Based on Leo's answer but I

• removed one column.
• replaced the \big) with a vertical rule.
• made all rows have equal width.

\documentclass{article}
\usepackage{amssymb}
\usepackage{array}

\newcommand{\x}[1]{\multicolumn{1}{|r}{#1}}

\begin{document}

\[
x^3 - x + 1 = (x-1)(x^2+x) + 1 \in \mathbb{F}_3[x]
\]

\[
\renewcommand\arraystretch{1.2}% specify the vertical stretch
\begin{array}{*{6}{>{\hfill}m{7mm}}}%specify the column width
&
    &
        &
            1&
                1&
                    0\\\cline{3-6}
1&
    -1&
        \x{1}&
                0&
                    -1&
                        1\\
&
    & 
        1&
            -1&
                &
                    \\\cline{3-5}
&
    &
        &
            1&
                -1&
                    \\
&
    &
        &
            1&
                -1&
                    \\\cline{4-6}
&
    &
        &
            &
                &
                    1\\
\end{array}
\]

\end{document}