Math Help Forum - View Single Post - Prove that there is no complex root such that both the Re and Im parts are rational.

mathwizard · #1 June 18th, 2008, 05:42 PM

Let $P$ be a polynomial with integer (real) coefficients such that; (1) - the coefficient of the leading term and the coefficient of the independent term (counted as the coefficient of $x^0, 0$ degree) are odd; (2) -The total number of odd coefficients is odd.

Like in
$P(x) = x^3 - 5x^2 + 2x -7$
$P(x) = 9x^3 - 6 x^2 + 3x -5$
$P(x) = x^4 +5x^3 + 7x^2 + x +1$
$P(x) = 7x^5 + 2x^4 - x^3 + 2x^2 - 8x -3$
Prove that $P$ has no root such that both the real and the imaginary parts are rational. In other words, if $a + bi$ is a root of $P$ , then at least one of the numbers $a$ and $b$ is irrational