find multiplicative inverses of the given elements in the gi

find multiplicative inverses of the given elements in the given fields:

Solution

1) Since any element in Q[x]/<x^2-2> can be uniquely written as [m + nx] for some m, n in R,
we want to find [m + nx] such that [a+bx] [m+nx] = [1].

==> [am + (an + bm)x + bnx^2] = [1]
==> [am + (an + bm)x + bn * -1] = [1], since [x^2 - 2] = [0] ==> [x^2] = [2]
==> [(am - bn) + (an + bm)x] = [2]

So, we need am - bn = 1 and bm + an = 0.
==> m = a/(a^2 + b^2) and n = -b/(a^2 + b^2).

Hence, [a + bx]^(-2) = [a/(a^2 + b^2) - bx/(a^2 + b^2)].
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