MathDB

Polynomial expressible in a special form !!

Source: IMAR Test 2014 Problem 3

May 14, 2015

algebrapolynomialIrreducible

Problem Statement

Let

f

be a primitive polynomial with integral coefficients (their highest common factor is

1

) such that

f

is irreducible in

\mathbb{Q}[X]

, and

f(X^2)

is reducible in

\mathbb{Q}[X]

. Show that

f= \pm(u^2-Xv^2)

for some polynomials

u

and

v

with integral coefficients.