Theorem 1

Let D be an R-algebra of finite degree that is a field. Then D is isomorphic to R, C, or H.

Denote the center of D by Z, and let L be a maximal commutative subfield of D. We have $[D : Z] = [L : Z]^2$ by VIII, p. 265, Corollary 2; we also have $[L : R]\leqslant 2$ because C is an algebraic closure of R. There are consequently three possible cases:

a) We have R = Z = L, so [D : Z] = 1 and D = R.

b) We have $R\not= Z$ and Z = L, so [D : Z] = 1 and D = L. In this case, D is isomorphic to C.

c) We have R = Z and [L : R] = 2, so [D : R] = 4. By Proposition 4 of VIII, p. 364, the R-algebra D is isomorphic to a quaternion algebra of type $(\alpha ,0, \gamma )$, where $\alpha$ and $\gamma$ are nonzero elements of R. Let $i\in D$- Z be such that $i^2=\alpha$. We have $\alpha \not= 0$. If $\alpha >0$, then there exists an $a\in R$ such that $a^2=\alpha$ (VI, §2, No. 6, p. 26, Theorem 3); we then have $(a-i)(a+i) = 0$, which is absurd because D is a field. So we have $\alpha <0$. The inequality $\gamma <0$ is shown analogously. There then exist elements $a$ and $c$ of $R^*$ such that $\alpha =-a^2$ and $\gamma =-c^2$ (loc. cit.). The algebra D is therefore isomorphic to the quaternion algebra of type $(-1,0,-1)$ (VIII, p. 362, Remark 2), that is, to H.