Is there a name for a polynomial with (real) integer coefficients, and for which all of the roots are Gaussian integers (i.e. both real and imaginary parts are integers)?
what might be a test for whether a given polynomial with (real) integer coefficients has all roots being Gaussian integers, other than brute force finding all the roots?
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3 Answers
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For 1, I would call such a polynomial "an integral polynomial that splits completely over the Gaussian integers". I don't think these polynomials have any specific name.
For 2, the rational root test works here just as well. It suffices to factor the constant coefficient of the polynomial (over the Gaussian integers), and to check whether these factors give you all roots.
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1. No, there isn't any name for such polynomials.
2. Polynomial functions are smooth. Therefore, they have derivatives at all points. So, one thing you could do is determine the minima and/or maxima. After determining them, you must check whether the function will touch the x-axis. You can use the Ehrlich-Aberth method to find the roots. Perhaps, you could refer to this post.
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One thing to note is that since $\mathbb Z$ and $\mathbb Z[i]$ are the algebraic integers of $\mathbb Q$ and $\mathbb Q(i)$, respectively, the following are equivalent for any integer polynomial $f$:
All roots of $f$ are Gaussian integers.
$f$ splits in $\mathbb Q(i)$, and $f$ divided by its content is monic, i.e., the leading coefficient divides all other coefficients.
