From A Course in Algebra by Vinberg (p. 13) :

The field of complex numbers is a field $\mathbb{C}$ which

(i) contains $\mathbb{R}$ as a subfield;

(ii) contains an element $i$ such that $i^2 = -1$;

(iii) is minimal among the fields with properties (i) and (ii), i.e., if $K$ is a subfield of $\mathbb{C}$ containing $\mathbb{R}$ and $i$, then $K = \mathbb{C}$.

Question. Why are the two properties listed in condition (iii) equivalent?

The first says that if $K$ is a field containing $\mathbb{R}$ and $i$, then $\mathbb{C} \subseteq K$. Therefore if $K$ is also a subfield of $\mathbb{C}$, then $K = \mathbb{C}$. So the first property implies the second. But I don't see how the second implies the first.

  • 2
    $\begingroup$ The sentence after 'i.e.' is his definition of "minimal". There is nothing to prove. $\endgroup$ – lhf Dec 8 '16 at 12:05

The minimality follows from the degree theorem. If $K$ is an intermediate field, we have $$ 2=[\mathbb{C}:\mathbb{R}]=[\mathbb{C}:K]\cdot [K:\mathbb{R}] $$ so that one of the two degrees has to be $1$, i.e., $K=\mathbb{R}$ or $K=\mathbb{C}$.


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