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Let $f(x_1,...,x_n)\in \mathbb R[x_1,...,x_n]$ be an irreducible (over $\mathbb R$) polynomial with real coefficients. Let $f$ has a factor in $ \mathbb C[x_1,...,x_n]$ of degree $\geq 1$.

Is $f$ of the form $f=\alpha r\cdot c(r)$, where $\alpha \in \mathbb R^*$, $r(x_1,...,x_n)$ is a some polynomial with $\mathbb C[x_1,...,x_n]$ and $c(r)$ is a polynomial obtained from $r$ by replacing its coefficients by their complex conjugate numbers?

Alex
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1 Answers1

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Let $r|f$ in $\mathbb C[x_1,...,x_n]$ and is irreducible over $\mathbb C$. Then also $(cr)|f$ and is irreducible over $\mathbb C$, moreover $r\cdot (cr) \in \mathbb R[x_1,...,x_n]$ and $r, cr$ are relatively prime. Hence $r(cr)|f$ and $rc(r)\in \mathbb R [x_1,...,x_n] $. Since $f$ is irreducible over $\mathbb R$, then $f=\alpha r (cr)$, with some nonzero real constant $\alpha$.

A.B
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