Existence of irreducible polynomial

Question

I am looking for irreducible polynomials in $\mathbb{F}_{11}$ with the the form h(y) = $y^7+$. My considerations are: the group order is 10 and is relatively prime to the degree 7 of the polynomial. Therefore no irreducible polynomials exist of this form. Is this correct ?

Answer 1

Your argument is correct only for showing that there is no irreducible polynomial of the form $X^7 + a$ with $a$ in the field. If this is the form you are looking for it is correct. However, not at all each degree seven polynomial is of that form, so that you can not conclude that there are no irreducible polynomials of degree seven.