Division Ring is Vector Space over Prime Subfield
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Theorem
Let $\left({K, +, \times}\right)$ be a division ring.
Let $\left({S, +, \times}\right)$ be the prime subfield of $K$
Then $\left({K, +, \times_S}\right)_S$ is an $S$-vector space, where $\times_S$ is the restriction of $\times$ to $S \times K$.
Proof
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Sources
- Seth Warner: Modern Algebra (1965): $\S 26$: Example $26.2$
