This came up in my logic course.(adsbygoogle = window.adsbygoogle || []).push({});

The professor writes that in GF(2), the polynomials [tex]3xy^5[/tex] and [tex]\frac{1}{2}x^2y[/tex] respectively can be reduced to [tex]3xy[/tex] and [tex]xy[/tex].

I understand that [tex]y^5=(y^2)(y^2)y=(1)(1)y=y[/tex], but also in GF(2), for any x, we have x+x=0. So it seems to me that [tex]3xy^5[/tex] can be further reduced to just [tex]xy[/tex] because we have [tex]3xy=xy+xy+xy=0+xy=xy[/tex].

For the other, I am clueless. Sure, [tex]x^2y=xy[/tex] because in GF(2), for any x, we have x²=x. But what happened to the 1/2? Worse, what is 1/2? It's the inverse of 2. And 2 is 1+1. But 1+1=0, and 0 has no inverse in a field.

What's going on here?

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# Coding and GF(2)

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