Is the splitting field of $x^7+x^5+x^2+1$ over the field $F$ with 25 elements a separable extension?
I know that $F\simeq \mathbb{Z}_{25}$. I found the roots in $\mathbb{Z}_{25}$ with a calculator: $4,7,9,14,19,24.$
I know it looks like I'm just asking for the answer, but I'm just learning separable extensions, and yet everything seems very awkward to me. What can I do to see if this extension is separable? Does exists any method, like step-by-step?
The field $\Bbb F_{25}$ with $25$ elements is a quadratic extension of $\Bbb F_5=\Bbb Z_5$. As such, any factoring of a polynomial over $\Bbb F_5$ is valid also over $\Bbb F_{25}$. And over $\Bbb F_5$ we have $$ x^7+x^5+x^2+1=(x^2+1)(x^5+1)\\ =(x^2-4)(x+1)^5=(x+2)(x-2)(x+1)^5 $$ Which is to say the polynomial splits completely over $\Bbb F_5$ and therefore also over $\Bbb F_{25}$. And it is not difficult to see that $\Bbb F_{25}/\Bbb F_{25}$ is a separable extension.