In Q[t], define the equivalence relation ~ as f(t) ~ g(t) precisely when f(t) - g(t) is a multiple of t^2 - 5. We define the addition and multiplication of equivalence classes as [f(t)] + [g(t)] = [f(t) + g(t)] and [f(t)] * [g(t)] = [f(t) * g(t)](adsbygoogle = window.adsbygoogle || []).push({});

(Assume: ~ is an equivalence relation, Addition/Mutliplication of equivalence classes is well-defined, and every equivalence class contains exactly one element of the form a + bt, where a, b in Q)

a) Find a, b in Q such that [3t^3 - 5t^2 + 8t - 9] = [a + bt]

b) Find a, b in Q such that [2t + 7] * [7t + 11] = [a + bt]

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

d) Find a, b in Q such that [a + bt]^2 + [-2][a + bt] = [19] (Two possible answers)

e) Find a, b in Q such that [2 - t] * [a + bt] = [1]

f) Which equivalence classes are zero divisors?

g) Which equivalence classes have multiplicative inverses?

h) How many equivalence classes are there whose square is equal to [6] ?

Note: * means multiplication

Now, I think I found the following solutions. Can someone verifty these and help solve the remaining parts?

a) [35t - 54]

b) [71t + 147]

c) ??

d) ??

e) ??

f) [0] = [t^2 - 5] (are these the only zero divisor classes??)

g) ??

h) ??

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Equiv. Class Problem

**Physics Forums | Science Articles, Homework Help, Discussion**