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I've tested polynomials of degree 6 with random integer coefficients $|a_i|<50$ in test-series of $10,000$. The probability of a random primitive polynomial of the kind to be reducible seems to be a little more than $2 \%$.

For sums of three random irreducible polynomials, primitive sum or not, the probability of being reducible also seems more than $2 \%$.

However, for sums of three random Eisenstein polynomials, primitive sum or not, the probability of being reducible seems less than $1 \%$.

Other degrees and limits for the coefficients gives just about the same results.

How to explain this property of this class of polynomials?

Hint? A perhaps even more peculiar property is that for the sum of two Eisenstein polynomials, the probability of being reducible is about $70 \%$! While corresponding test for two random irreducible polynomials gives about $2 \%$ reducible.

The main reason why sums of two Eisenstein polynomials has high probability of being reducible seems to be that all coefficients tends to be even. When such a sum is divided with the greatest common divisor about $5\%$ are reducible.

Using an Eisenstein polynomial twice, getting polynomials of the form $2p+q$ gives about $1.3 \%$ reducible.


A random Eisenstein polynomial is here a random polynomial of degree 6 divided with the greatest common divisor of the coefficients, which is selected if it satisfies the Eisenstein criteria. And similar for a random irreducible polynomial. Just brutal force trial and error.

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  • $\begingroup$ By Eisenstein polynomial, do you mean a polynomial that Eisenstein's criterion shows to be irreducible? Assuming so, have you been summing those polynomials that are shown to be irreducible modulo the same prime, or have you been checking, say, the sum of a polynomial shown to be irreducible by Eisenstein's criterion with $p=3$ with another shown by $p=5$? $\endgroup$ – Carl Schildkraut Feb 13 at 7:39
  • $\begingroup$ @CarlSchildkraut: see addendum. $\endgroup$ – Lehs Feb 13 at 11:15

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