If $\mathbb{K}$ is an extension of $\mathbb{Q}$ of degree $15$, which of these elements cannot belong to $\mathbb{K}$?

Question

If $\mathbb{K}$ is an extension of $\mathbb{Q}$ of degree $15$, which of these elements cannot belong to $\mathbb{K}$ ?

I know that it has to do with minimal polynomial but I am unsure of how to determine them.

Answer 1

The minimal polynomials are (A) $x^2+144$, (B) $x^{15}-3$, (C) $x^3-7$, and (D) $x^5-10$,

of degrees $2$, $15$, $3$, and $5$, respectively.

If $\alpha\in \Bbb K$, then $\mathbb Q(\alpha)$ is a subfield of $\mathbb K$ and therefore its degree is a factor of $[\mathbb K:\mathbb Q]$.

Only one of these degrees is not a factor of $15$, and that is the element that cannot belong to $\mathbb K$.