Coding & GF(2): Understanding Logic in a Field

[quasar987]

This came up in my logic course.

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

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

For the other, I am clueless. Sure, $$x^2y=xy$$ 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?

 

[Hurkyl]

First off, just to make sure it's clear, we're talking about polynomial functions, not polynomials. The distinction is subtle, yet vital.

Secondly, $x^2 \not\equiv 1$; I'm not sure where you got that idea. The right fact is $x^2 \equiv x$.

Thirdly, 1/2 doesn't make any sense in GF(2), because as you say, it's equivalent to 1/0.