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) ??

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# Equiv. Class Problem

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