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.