Prove that $7^{\frac 14}$ is not rational using the Unique Factorization Theorem.

Question

I am currently trying to prove this using the Unique Factorization Theorem and I am stuck.

I attempt to prove this BWOC and assume $7^{\frac 14}$ is rational so that it can be expressed as $\frac ab$.

Thus $7 = \frac {a^4}{b^4} \implies 7b^4 = a^4$. If both $a^4$ and $b^4$ have unique prime factorizations, how do I get there to conclude that $7^{\frac 14}$ is irrational?

Answer 1

First you can assume that $\text{gcd}(a,b) = 1$. Using UFT for $a,b$ you can argue that $a = 7^{x_1}\cdots a_n^{x_n}, b = 7^{y_1}\cdots b_n^{y_n}$. Thus $\text{gcd}(a,b) > 1$, contradiction.

Answer 2

Alternative approach:

Suppose that $a= 7^{\frac 1 4}$ is rational.

Then, $a^4=7$, so $a$ is a root of $X^4-7$. But the rational root theorem says it's not.