Exercise in arithmetic of a finite field

Question

I am in difficult in resolving this exercise in Galois Theory : "in $GF(2^5)$ calculates the product $(1,1,1,0,1)(0,1,0,1,0)$ , generator of $GF(2^5)^*$ ". I don't know how to proceed.. thank you

Answer 1

Hint: Do you know what the multiplication by $(0, 1, 0, 1, 0) = (0, 1, 0, 0, 0) + (0, 0, 0, 1, 0)$ is given as a matrix with coefficients in $\mathbb{F}_2$? From this it should be fairly easy.

Answer 2

I just know that $(0,1,0,1,0)$ represent $t^3+t$ polynomial .. Ps: I m new here, is it possible upload image ? And how to write correctly the mathematical formulas?