I've started studing fields and their extensions. As an exercise I proved that $[\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}]=4$ by showing that $B=(1,\sqrt2,\sqrt3,\sqrt6)$ is a base for the extension field as a vector space over $\mathbb{Q}$.
Now, I couldn't of not noticed the analogy between this ring and the quaternion ring. Actually it feels more like there is an anolgy between how $\mathbb{Q}(\sqrt2,\sqrt3)$ relate to $\mathbb{Q}$ and how the quaternion ring relates to $\mathbb{R}$. Indeed: $\sqrt2^2,\sqrt3^2,\sqrt6^2\in 1\mathbb{Q}$ and $\sqrt2\cdot\sqrt3\in\sqrt6\mathbb{Q}, \sqrt2\cdot\sqrt6\in\sqrt3\mathbb{Q}, \sqrt3\cdot\sqrt6\in\sqrt2\mathbb{Q}$ - similar to the quaternions.
By this analogy it seems like the linear independence of $B$ is implied by the fact that the quaternions are a ring and hence have a unique $0$ element. Showing that $Sp(B)=\mathbb{Q}(\sqrt2,\sqrt3)$ becomes very easy by the fact that the quaternions are a division ring.
How would you formally express this analogy (if you agree that is a good analogy). Clearly the same idea can be applied to the how the complex field relates to the reals and how $\mathbb{Q}(\sqrt2)$ related to $\mathbb{Q}$ Thanks