Equivalence Relations: In Q[t], define ~ as f(t) ~ g(t) when

[kbm2122]

In Q[t], define the equivalence relation as ~ as f(t) ~ g(t) when f(t) - g(t) is a multiple of t^2 - 5

Addition and multiplication is well defined as [f(t)] + or * [g(t)] = [f(t) + or * g(t)]

1) Which equivalence classes have zero divisors?

2) Find two equivalence classes whose square is equal to [5]

 

[daon]

Re: Equivalence Relations

In Q[t], define the equivalence relation as ~ as f(t) ~ g(t) when f(t) - g(t) is a multiple of t^2 - 5

Addition and multiplication is well defined as [f(t)] + or * [g(t)] = [f(t) + or * g(t)]

1) Which equivalence classes have zero divisors?

2) Find two equivalence classes whose square is equal to [5]

For 2,

You know: f~g when f-g = k(t^2-5).

If f~g, then [f]=[g], so [f-g] = [0] = k[t^2-5], for some k.

Then [1/k*(g-f)+t^2)] = [t^2] = [5], so: [sqrt(1/k*(g-f)+t^2)]^2 = [t]^2 = [5]

I'm not too sure about my answer/method, so wait until someone else stops by.