Find the multiplicative inverse of 1+3√2 in the ring Q(√2) and use it to solve the equation (1+3√2)x=1−5√2.
I think that the inverse is the conjugate, so it would be 1−3√2, but then I don't know where to use in the equation that needs to be solved.
Answer
Let a+b√2∈Q(√2) be the inverse of 1+3√2, i.e. (a+b√2)(1+3√2)=1. Then
1=a+6b+(3a+b)√2,
i.e.
3a+b=0, a+6b=1.
It follows that
a=−117, b=317.
Now
(1+3√2)x=1−5√2⟺x=(1+3√2)−1(1−5√2),
i.e.
x=117(−1+3√2)(1−5√2)=−3117+817√2.
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