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We can solve a system of equations over reals using Mathematica as follows.

Solve [x^2+y^2+z^2==1 && x+y==1, {x,y,z}]

How one can solve a system of equations over a finite field like GF(2) using Mathematica?

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up vote 6 down vote accepted

Solve[x^2+y^2+z^2==1 && x+y==1, {x,y,z}, Modulus->2]

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Alternatively, Solve[{x^2 + y^2 + z^2 == 1, x + y == 1}, {x, y, z}, Modulus -> 2]. –  J. M. Dec 1 '11 at 12:16
    
This is fine, if you work over the prime field. But has Wolfram implemented support for other finite fields? I asked around about that may be 15 years ago (an am still running Mathematica ver 3.0 :-) –  Jyrki Lahtonen Dec 4 '11 at 15:14
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I think so: Modulus->8 works. –  user7530 Dec 4 '11 at 16:20
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