  A polynomial f(t) with rational coefficients is strongly irreducible if
f(t^k) is irreducible for all positive integers k. Likewise, two polynomials f
and g are strongly coprime if f(t^k) and g(t^l) are relatively prime for all
positive integers k and l. We provide some sufficient conditions for strong
irreducibility and prove that the Alexander polynomials of twist knots are
pairwise strongly coprime and that most of them are strongly irreducible. We
apply these results to describe the structure of the subgroup of the rational
knot concordance group generated by the twist knots and to provide an explicit
set of knots which represent linearly independent elements deep in the solvable
filtration of the knot concordance group.
