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Over $GF(p^m)$ there are exactly $φ(p^m − 1)/m$ primitive polynomials of degree $m$. I'm wondering if a fast generator exists. All of the primitive polynomials can be expressed as a number, so perhaps there good filters in number theory. Is the number $p^m − 1$ useful as one of the filters?

For $GF(2^9)$, 48 primitive polynomials, first is $x^9+x^4+1$, $(529, 539, 545, 557, 563, 601, 607, 617, 623, 631, 637, 647, 661, 675, 677, 687, 695, 701, 719, 721, 731, 757, 761, 787, 789, 799, 803, 817, 827, 847, 859, 865, 875, 877, 883, 895, 901, 911, 949, 953, 967, 971, 973, 981, 985, 995, 1001, 1019)$

For $GF(3^6)$, 48 primitive polynomials, first is $x^6+x+2$, $(734, 737, 761, 791, 833, 836, 845, 854, 869, 878, 905, 908, 932, 962, 974, 1001, 1013, 1022, 1046, 1070, 1073, 1085, 1094, 1106, 1133, 1145, 1160, 1166, 1172, 1184, 1208, 1217, 1262, 1271, 1286, 1295, 1310, 1316, 1346, 1358, 1367, 1373, 1388, 1400, 1424, 1439, 1442, 1457)$ All $(2)$ mod(3).

For $GF(5^4)$, 48 primitive polynomials, first is $x^4+x^2+2 x+2$, $(662, 663, 667, 668, 732, 733, 747, 748, 758, 763, 767, 772, 783, 802, 807, 828, 847, 873, 883, 887, 893, 897, 913, 928, 937, 952, 967, 993, 1007, 1013, 1017, 1023, 1043, 1053, 1067, 1077, 1087, 1113, 1132, 1137, 1143, 1148, 1173, 1177, 1197, 1203, 1207, 1233)$ All $(2, 3)$ mod(5).

For $GF(7^3)$, 36 primitive polynomials, first is $x^3+3 x+2$, $(366, 380, 387, 401, 410, 429, 431, 452, 471, 485, 487, 494, 501, 506, 515, 527, 529, 534, 555, 562, 564, 571, 592, 606, 611, 613, 618, 625, 634, 641, 646, 660, 669, 676, 681, 683)$ All $(2,4)$ mod(7).

Are there fast generators for these number lists that could also generate the number lists for things like $GF(2^{14})$, $GF(19^{3})$, $GF(3^{10})$, $GF(31^3)$, $GF(79^4)$, and $GF(83^4)$? Barring that, are there good filters for reducing the number of polynomials that need to be checked?

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    Find $f$ one irreducible polynomial of degree $n$ over $\mathbb{F}_p$, find a generator $a$ of $\mathbb{F}_{p^n}^\times = (\mathbb{F}_p[x]/(f))^\times$, generate (with the Galois conjugates) the minimal polynomial of $a^m$ for $gcd(m,p^n-1) = 1$. – 2017-02-16
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    What user1952009 suggests is probably good. When we are in Mersenne prime case, $2^m-1$ will be a prime, and the list of primitive polynomials has exponential length (may be it has in the generic case also, and the Mersenne case is just an extremal example). – 2017-02-17

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