# Symbolic computations in field extensions with Mathematica

In Mathematica I would like to make symbolic computations in field extensions. For example, if $a$ is a root of $x^3+3x-1$, I would like to get $$a^4 = a-3a^2$$ if I asked the machine to compute $a^4$. How can I do this? I tried

a^3+3a-1=0;
Expand[a^4]


but it does not work and just outputs $a^4$.

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Do you want to solve the equation first and then find $a^4$ which $a$ is any root? –  B. S. Mar 24 '13 at 12:06
Using substitutions as here math.stackexchange.com/questions/48838/… should work. –  Andrew Mar 24 '13 at 12:07
Yea with $a$ being any root @BabakS. I would like to get $a-3a^2$ as output. –  Heitor Fontana Mar 24 '13 at 12:17
@Andrew : Thanks, I took a look at the link but it's not clear to me. So what would you write in my case? –  Heitor Fontana Mar 24 '13 at 12:18
Have you tried PolynomialMod? –  Elmar Zander Mar 24 '13 at 12:20