I was told that sometimes in characteristic 2 that $X^n + X + 1$ is reducible mod 2. What is the smallest $n$ where that is true?
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Tom Lewia
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1Have you searched for such an $n$? What have you found? – abiessu May 18 '16 at 15:22
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The simplest approach is simply to list all polynomials mod 2 of orders 1,2,3 noting which are irreducible (there are only five such). Now check to see if $1+x+x^4,1+x+x^5,1+x+x^6$ are reducible (if they are then one of the five must be a factor). – almagest May 18 '16 at 15:29
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The smallest is $n=1$... Maybe you should add $n\geq2$? – Johannes Huisman May 18 '16 at 15:53
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@JohannesHuisman Is $1+2x=1$ reducible? – almagest May 18 '16 at 15:55
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See http://math.stackexchange.com/questions/32197/find-all-irreducible-monic-polynomials-in-mathbbz-2x-with-degree-equal. – lhf May 18 '16 at 15:59
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@almagest: yes, $1$ is reducible. In an integral domain $A$ an element $a$ is irreducible if 1. it is different from $0$ and is not a unit, and 2. each time one can write $a=bc$ in $A$ then either $b$ or $c$ is a unit in $A$. – Johannes Huisman May 18 '16 at 18:36
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Observe that $x^3+1=(x^2+x+1)(x+1)$, and therefore $x^{3k}+1$ is divisible by $x^2+x+1$. Hence so is $x^{3k+2}+x^2$. This implies that if $n>2$ and $n\equiv2\pmod3$ then $x^n+x+1$ is divisible by $x^2+x+1$. This doesn't account for all the entries in Johannes' list, but is something you can work out by hand. – Jyrki Lahtonen May 18 '16 at 22:38
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Well, that was easy: $n=5$ is the smallest value of $n$ for which $x^n+x+1$ is reducible. Indeed, $x^5+x+1=(x^2+x+1)(x^3+x^2+1)$. The other values for wich $x^n+x+1$ is reducible in $\mathbf F_2[x]$ are, for $n<100$:
8,
10,
11,
12,
13,
14,
16,
17,
18,
19,
20,
21,
23,
24,
25,
26,
27,
29,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44,
45,
47,
48,
49,
50,
51,
52,
53,
54,
55,
56,
57,
58,
59,
61,
62,
64,
65,
66,
67,
68,
69,
70,
71,
72,
73,
74,
75,
76,
77,
78,
79,
80,
81,
82,
83,
84,
85,
86,
87,
88,
89,
90,
91,
92,
93,
94,
95,
96,
97,
98,
99
I determined this with a little sage program:
A.<x>=PolynomialRing(GF(2))
for i in range(2,100):
if not((x^i+x+1).is_irreducible()):
print i
Johannes Huisman
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