irreducible and minimum stuff

Find an irreducible poly that is not a min poly. PhL 6.30.13 This led to my picture and discussion of the classification of irreducible polys of GF(q). It is now installed in Galois doc near Fig 5.5 (1) I wrote some code to attempt to factor all polynomials for GF(24) which have the standard form ending in a 1. Here it is. There are 16 of interest, but the first two are trivial. Here are the possible categories factors into all (x-a) factors where in all factors a ≠ 1 irreducible and min poly factors in this way, but has at least one (x+1) factor reducible hence not min poly has a single but non-linear in x factor irreducible and not min poly has multiple factors with at least one non-linear reducible hence not min poly As I scan down the list, I seem to find certain ones that don't factor. Here they are: 1 + x + x3 1 + x2 + x3 These two are irreducible in R because they cannot be factored! Since they cannot be factors even in GF(q), they are not min polys. (2) Here is a list of all min polys for GF(24). p1(x) = (x - α)(x - α2)(x - α4) (x - α8) = x4 + x + 1 p7(x) = (x - α7)(x - α14)(x - α13)(x - α11) = x4 + x3 + 1 m3(x) = (x - α3)(x - α6)(x - α12)(x - α9) = x4 + x3 + x2 + x + 1 m5(x) = (x - α5)(x - α10) = x2 + x + 1 (6.21) Therefore, the two polys above are irreducible but they are not min polys! So I now have an example of a polynomial which is irreducible but not a min poly. (3) Question: are all irreducible polys of full degree k also min polys? That was the case in the above example. We can certainly have degree k polys which are reducible, as example above shows. I looked at the case 27 and the same seems true. You have to find a single factor containing a poly of degree 7 which is non-linear. Here is the answer: Fact 14: If monic irreducible f(x) is used to construct a representation of GF(q), then f(x) is a minimum polynomial of GF(q) in that representation. (5.34) To this I could add some Corollaries: Corollary: If monic irreducible f(x) is of degree m, then it is a minimum polynomial for GF(pm). (5.34a) Corollary: If monic irreducible f(x) is not a minimum polynomial for GF(pm), it must have degree < m. (5.34b) Here then is the picture I have been waiting for An example over on the right for GF(24) is 1 + x + x3 . Next question: How show all those order-reversal things. New doc.