In this paper, for any given monic cubic irreducible polynomial f is an element of & Qopf;[x], we describe the conditions on rational numbers a not equal 0 and b, c under which there exist g, h is an element of & Qopf;[x] such that f (ax 2 + bx + c) = g(x)2 + h(x)2 . In particular, our results imply that for any such f there is a suitable triplet (a, b, c), where a is an element of N, b = 0 and c is an element of & Qopf;, with such a representation. However, this is not always the case with integers a not equal 0 and b, c. In this direction, we show that, unlike it has been previously conjectured, there exist cubic monic irreducible polynomials in & Zopf;[x], say, f (x) = x 3 - 2 or f (x) = x 3 + 3, such that for any integers a not equal 0 and b, c the polynomial f (ax 2 + bx + c) is not expressible as a sum of squares of two polynomials in & Qopf;[x].
