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Bruno J. · #4 June 17th, 2009, 05:01 PM

Here is another simple proof of the nice theorem (1) :

Denote $\overline{f(x)}$ the complex conjugate of $f(x)$ . Then for any two functions $f(x), g(x) \in \mathbb{C}[x]$ we have $\overline{f(x)g(x)}=\overline{f(x)}\: \: \overline{g(x)}$ .
This follows directly from the similar property of conjugation for elements of $\mathbb{C}$ and from the definition of a polynomial product.

Then $\overline{f(x)\overline{f(x)}}$ = $\overline{f(x)}\: \: \overline{\overline{f(x)}} = \overline{f(x)}f(x) = f(x)\overline{f(x)}$ .

Hence $f(x)\overline{f(x)}$ is its own conjugate, i.e. $f(x)\overline{f(x)} \in \mathbb{R}[x]$ .