TITLE: Multiplicative inverse of a quadratic algebraic number $\,a+b\sqrt 2$
QUESTION [5 upvotes]: Find the multiplicative inverse of $1+ 3\sqrt{2}$ in the ring $\mathbb{Q}(\sqrt{2})$ and use it to solve the equation $(1+3\sqrt{2})x=1-5\sqrt{2}$.
I think that the inverse is the conjugate, so it would be $1-3\sqrt{2}$, but then I don't know where to use in the equation that needs to be solved.

REPLY [2 votes]: $\rm\alpha\!\neq\! 0\Rightarrow  0\!\ne\!\alpha\bar\alpha\! =\! n\in \Bbb Z,\,$ so $\,\bar \alpha\,$ times $\rm\: \alpha\, x = \beta\,$ $\rm\Rightarrow\, n\, x = \bar\alpha\beta, \:$ i.e. in fraction language $$\rm\: x =  \dfrac{\beta}{\color{#c00}{\alpha}} = \dfrac{\bar\alpha\beta}{\bar\alpha\alpha} = \dfrac{\bar\alpha\beta}{\color{#0a0}n}$$
This is known as rationalizing the denominator. The key idea is that  every irrational algebraic $\,\alpha\,$ has a "simpler" (i.e. rational) multiple, its norm $\,\color{#0a0}{\rm n} = \alpha\bar \alpha,\,$  so we can reduce division by an irrational $\,\color{#c00}{\alpha}$ to division by the rational $\,\color{#0a0}{\rm n},\,$ which is simpler. This is a special case of the general method of simpler multiples.