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A polynomial f(t) with rational coefficients is strongly irreducible iff(t^k) is irreducible for all positive integers k. Likewise, two polynomials fand g are strongly coprime if f(t^k) and g(t^l) are relatively prime for allpositive integers k and l. We provide some sufficient conditions for strongirreducibility and prove that the Alexander polynomials of twist knots arepairwise strongly coprime and that most of them are strongly irreducible. Weapply these results to describe the structure of the subgroup of the rationalknot concordance group generated by the twist knots and to provide an explicitset of knots which represent linearly independent elements deep in the solvablefiltration of the knot concordance group.

Strong coprimality and strong irreducibility of Alexander polynomials