Arithmetic in Finite Fields

This calculator can be used to perform computations in finite fields. By selecting a prime number $p \le 541$ and an exponent $n \le 10$, you define that calculations are carried out in the field $\mathbb{F}_{p^n}$ with $p^n$ elements.

Input is entered via the keypad. The result appears only after pressing the $\fbox{$=$}$ key. The key functions are briefly explained below:

$\fbox{$\Box^m$}$ activates exponent input: the digits entered afterward are interpreted as a natural-number exponent.
$\fbox{←}$ deletes the most recently entered character.
$\fbox{ ⟳$^{\phantom{.}^{\phantom{.}}}$}$ clears the entire input line.
After displaying a result with $\fbox{$=$}$, you can continue calculating immediately: if the next key is $\fbox{$+$}$, $\fbox{$-$}$, $\fbox{$\times$}$, or $\fbox{$\div$}$, the calculator continues with the last result. With any other key, a new input starts.
$\fbox{\textsf{last}}$ recalls the most recently executed calculation.
$\fbox{$\in$ \textsf{F}}$ displays the elements of $\mathbb{F}_{p^n}$.
$\fbox{\textsf{Mod}}$ displays the generating polynomial (only for $n > 1$).
Decimal numbers with finitely many digits after the decimal point are allowed as input. Internally, they are interpreted as rational numbers and evaluated as division in $\mathbb{F}_{p^n}$.
To avoid confusion with the multiplication operator, polynomials are written in the indeterminate $y$.

This calculator was implemented by Ines Koob as part of her bachelor’s thesis. It is based on the open-source computer algebra system Sage.