Degree of a field extension

If $E$ is a field extension of a field $K$ , then $E$ can be viewed as a vector space over $K$ .

The degree of the extension $K\subseteq E$ , denoted as $[E:K]$ , is the dimension of $K$ viewed as a vector space over $K$ .

$\R\subseteq \mathbb{C}$ has degree 2, since $\{1,i\}$ forms a basis of $\mathbb{C}$ as a vector space over $\R$ .

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Let $E$ be a field extension of $\R$ of odd degree. Show that $E=\R$ .

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