Nature of algebraic structure

Question

I am given $G = \{x + y \sqrt7 \mid x^2 - 7y^2 = 1; x,y \in \mathbb Q\}$ and the task is to determine the nature of $(G, \cdot)$, where $\cdot$ is multiplication. I'm having trouble finding the inverse element (I have found the neutral and proven the associative rule.

Answer 1

For $a+b\sqrt{7}$ we seek $x+y\sqrt{7}$ such that $(a+b\sqrt{7})(x+y\sqrt{7})=1$. Expanding these brackets and comparing coefficients gives

$$\begin{align} ax+7by &= 1 \\ bx+ay &= 0 \end{align}$$

It is then just a task of solving for $x$ and $y$.

This can be done using matrices: $$\begin{align} \begin{pmatrix} a & 7b \\ b & a \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} &= \begin{pmatrix} 1 \\ 0\end{pmatrix} \\ \Rightarrow \begin{pmatrix} x \\ y \end{pmatrix} &= \begin{pmatrix} a & 7b \\ b & a \end{pmatrix}^{-1} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \end{align}$$

Note that $a^2-7b^2 \ne 0$ for any $a,b \in \mathbb{Q}$ and so this matrix is invertible. It comes out from here in a couple of lines; then $x+y\sqrt{7} = (a+b\sqrt{7})^{-1}$.

Answer 2

Here's a hint: try using the difference of squares formula on the left hand side of the equation $x^2 -7y^2 =1$.

Answer 3

\begin{eqnarray} \dfrac{1}{x+y\sqrt{7}} &=& \dfrac{x-y\sqrt{7}}{x^2-7y^2}\\ & =& x-y\sqrt{7} \end{eqnarray}

Answer 4

Hint. Your set has a certain formal similarity with $\{x+\sqrt{-1}y \mid x^2+y^2=1\}$ (arising by writing $-1$ for both occurrences of $7$), which is the unit circle in the complex plane. In the complex unit circle, the multiplicative inverse is just the complex conjugate, i.e. what you get by negating the $y$ coordinate.

It's worth a try to see if something similar works here.

Answer 5

Hint $\ \alpha\, \bar\alpha = 1\:\Rightarrow\: \bar\alpha = 1/\alpha$