Frobenius's Theorem/Lemma 3
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Lemma
Let $\struct {A, \oplus}$ be a quadratic real algebra.
Then:
- $A = \R \oplus U$
Proof
Let $a \in A \setminus \R$.
Then:
- $\exists \nu \in \R: a^2 - \nu a \in \R$
Therefore, if we set
- $u = a - \dfrac \nu 2 \in U$
then $u^2 = a^2 - \nu a + \nu^2/4 \in \R$, so
- $a = \dfrac \nu 2 + u \in \R \oplus U$
which proves the assertion.
$\blacksquare$