Rationals multiplying with irrationals giving 0?

Algebra Level 4

$\large a+b\sqrt[3]{2}+c\sqrt[3]{4} = 0$

Let $a$ , $b$ and $c$ be rational numbers satisfying the above equation. Find the number of such triples $(a, b, c)$ .

The answer is 1.

1 solution

Otto Bretscher
May 24, 2016

Let $p=\sqrt[3]{2}$ , with $p^2=\sqrt[3]{4}$ . The minimal polynomial of $p$ is $x^3-2$ , by Eisenstein, so that $f(p)\neq 0$ for any non-zero polynomial $f(x)=a+bx+cx^2$ with rational coefficiants $a,b,c$ . Thus $a=b=c=0$ ; the answer is $\boxed{1}$ .

Rationals multiplying with irrationals giving 0?

The answer is 1.

1 solution

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