Thanks, Mike! Looks like you tackled all the Galois fields GF(2^n), not just GF(2^(2^k)). How did you do it? Let me make sure I understand your notation. Does GF2_15_11 mean the polynomial 1 + x + x^15? I am surprised that the bit strings you found are so short (corresponding to most of the high-degree coefficients being zero). It makes me wonder whether you answered a different question than the one I asked, and computed the lexicographically first irreducible polynomial of degree n. That’s an interesting question too, but I see no reason why they should have the same answer. Jim Propp On Mon, Jan 18, 2021 at 3:35 PM Mike Speciner <ms@alum.mit.edu> wrote:
Here's the first 64, with the x^n term elided; constant term last. Not an obvious pattern, and, interpreted as a binary number, not in OEIS (either with or without the x^n term)...
'GF2_2_0' 'GF2_2_11' 'GF2_3_11' 'GF2_4_11' 'GF2_5_101' 'GF2_6_11' 'GF2_7_11' 'GF2_8_11011' 'GF2_9_11' 'GF2_10_1001' 'GF2_11_101' 'GF2_12_1001' 'GF2_13_11011' 'GF2_14_100001' 'GF2_15_11' 'GF2_16_101011' 'GF2_17_1001' 'GF2_18_1001' 'GF2_19_100111' 'GF2_20_1001' 'GF2_21_101' 'GF2_22_11' 'GF2_23_100001' 'GF2_24_11011' 'GF2_25_1001' 'GF2_26_11011' 'GF2_27_100111' 'GF2_28_11' 'GF2_29_101' 'GF2_30_11' 'GF2_31_1001' 'GF2_32_10001101' 'GF2_33_1001011' 'GF2_34_11011' 'GF2_35_101' 'GF2_36_110101' 'GF2_37_111111' 'GF2_38_1100011' 'GF2_39_10001' 'GF2_40_111001' 'GF2_41_1001' 'GF2_42_100111' 'GF2_43_1011001' 'GF2_44_100001' 'GF2_45_11011' 'GF2_46_11' 'GF2_47_100001' 'GF2_48_101101' 'GF2_49_1110001' 'GF2_50_11101' 'GF2_51_1001011' 'GF2_52_1001' 'GF2_53_1000111' 'GF2_54_1111101' 'GF2_55_1000111' 'GF2_56_10010101' 'GF2_57_10001' 'GF2_58_1100011' 'GF2_59_1111011' 'GF2_60_11' 'GF2_61_100111' 'GF2_62_1101001' 'GF2_63_11' 'GF2_64_11011'
On 18-Jan-21 12:01, James Propp wrote:
If I’m not mistaken, Conway’s notion of Nim-multiplication gives a way of singling out a “first” degree n irreducible polynomial over GF(2). For instance, x = *4 satisfies x^2 = *6, x^3 = *14, and x^4 = *5, so the first irreducible polynomial of degree 4 is x^4 + x + 1.
It would be interesting to peruse these polynomials for the first dozen powers of 2 to see if there’s a simple pattern.
When n isn’t a power of 2 you need to use Nim-values associated with infinite ordinals but that’s okay since they’re well-ordered.
Jim Propp
On Mon, Jan 18, 2021 at 7:23 AM Joerg Arndt <arndt@jjj.de> wrote:
Possibly making a fool of myself... Have you checked 'the Bible' (Lidl, Niederreiter)? I seem to recall that there is a nontrivial amount of coverage about binomial polynomials, specifically regarding irreducibility.
Best regards, jj
Let $n = 2^{e_0} q_1^{e_1} \cdots q_i^{e_i}$ be the prime factorization of $n$. Let $p$ be an odd prime. Is it true that $\exists k$ such that $x^n+k$ is irreducible over GF($p$) $\iff$ $(2^{min(e_0,2)} q_1 \cdots q_i) | (p-1)$ ?
Are there similar criteria for "larger" polynomials such as $x^n + k_1 x
* Mike Speciner <ms@alum.mit.edu> [Jan 14. 2021 17:11]: +
k_0$ ?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun