show that x^p - x + a is irreducible over GF(p) if a is different from 0

Show that x^{p} - x + a is irreducible over GF(p) if a is different from 0.

I assumed for contradiction that x^{p} - x + a is reducible. Then it must split into irreducible factors (none of them linear) over GF(p) or just Z_{p}. Say one of the irreducible factors has degree m < p. Then the size of its splitting field must be p^{m}. The splitting field of x^{p} - x + a consists of all the roots of that polynomial. Let b be a root of that polynomial. Then since the multiplicative group of the splitting field is cyclic, all roots of that polynomial can be expressed as some power of b. So I need to find the minimum polynomial of b. If the minimum polynomial of b is just x^{p} - x + a then I am done. So I assume the minimum polynomial of b is the irreducible factor with degree m < p that I mentioned above. Then the splitting field of the irreducible factor of degree m must be the same as the splitting field of x^{p} - x + a since they both contain all the roots of x^{p} - x + a.

I am having trouble finding a contradiction in this argument. Can someone help me with this proof? Thanks.

Re: show that x^p - x + a is irreducible over GF(p) if a is different from 0

suppose b is a root of .

let k be any element of .

then .

we then have p distinct roots of , so this IS the splitting field of .

now consider given by .

the assignment is a group homomorphism .

since , we know that or:

. the former implies that ,

that is, that b is in . since a ≠ 0, this cannot be true, so ,

which in turn means is irreducible over .

Re: show that x^p - x + a is irreducible over GF(p) if a is different from 0

why is or ? also, could you please explain your motivation for defining the group homomorphism from to ? i don't understand the reasoning for doing this.

Re: show that x^p - x + a is irreducible over GF(p) if a is different from 0

because is a homomorphsim, so it's image is a subgroup of the additive group of . but has prime order, so...

my motivation was to use the galois correspondence: .

Re: show that x^p - x + a is irreducible over GF(p) if a is different from 0

thanks for the reply. i understand that the image must be {0} or but what I am still trying to figure out is why does imply that ? I am having trouble seeing the connection between the degree of the field extension and the image of the map .

sorry i haven't learned galois theory yet so i was confused for a bit. but now that you posted the galois correspondence, it all makes sense now. thanks!