Joelly
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HOI
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An obvious "guess", practically forced by the way the problem is stated, is the function that maps $a+ b\sqrt{2}$ to $\begin{bmatrix} a & 2b \\ b & a \end{bmatrix}$. Is that an isomorphism? That is, is it "one to one", is it "onto", and does it "preserve the operation".
The first two are simple. To prove that this "preserves the operation" you need to take two members of the first set, $a+ b\sqrt{2}$ and $x+ y\sqrt{2}$, which map to $\begin{bmatrix}a & 2b \\ b & a\end{bmatrix}$ and $\begin{bmatrix}x & 2y \\ y & a\end{bmatrix}$ and show that the product $(a+ b\sqrt{2})(x+ y\sqrt{2})$ maps to the product $\begin{bmatrix}a & 2b \\ b & a \end{bmatrix}\begin{bmatrix}x & 2y \\ y & x\end{bmatrix}$.
Do those two multiplications and compare them.
To determine whether they are also isomorphic under addition, add those pairs and compare them.
The first two are simple. To prove that this "preserves the operation" you need to take two members of the first set, $a+ b\sqrt{2}$ and $x+ y\sqrt{2}$, which map to $\begin{bmatrix}a & 2b \\ b & a\end{bmatrix}$ and $\begin{bmatrix}x & 2y \\ y & a\end{bmatrix}$ and show that the product $(a+ b\sqrt{2})(x+ y\sqrt{2})$ maps to the product $\begin{bmatrix}a & 2b \\ b & a \end{bmatrix}\begin{bmatrix}x & 2y \\ y & x\end{bmatrix}$.
Do those two multiplications and compare them.
To determine whether they are also isomorphic under addition, add those pairs and compare them.
Joelly
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