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In a Mathematica session, I evaluate:

Needs["FiniteFields`"];
fld = GF[2];

Now, for example, I'd like to compute $a^3+a^5$ for $a \in{Z_2}$. The result I'd like to have is $0$, because in $Z_2$, $a^n=a$ and $a+a=0$.

I tried:

fld[{a^3}] + fld[{a^2}]

But Mathematica gives me:

Subscript[{Mod[a^2 + a^3, 2]}, 2]

How can I simplify $a^3+a^2$ in $Z_2$ to get zero?

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It would seem I gave you some bad advice. Daniel Lichtblau made some useful comments here. You should read them. –  m_goldberg May 22 at 23:59
1  
My advice conflicts in some ways with that of @m_goldberg. I do not claim that advice was bad-- for some purposes the FiniteFields package might in fact be what's needed. I only advocate avoiding it when there are alternatives, as it is unmaintained and moreover sometimes will do, in technical parlance, "weird things". –  Daniel Lichtblau May 23 at 0:09

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