Re: Is x^n + (x+2)^n irreducible over Q if n is a power of 2?

Peter L. Montgomery wrote:
In article <1157911324.533652.64880@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
"Edwin Clark" <eclark@xxxxxxxxxxxx> writes:
Is the polynomial x^n + (x+2)^n irreducible over Q if n is a power of
2?

It is true at least for n = 2^k, k=1..10.

Let f(x) be an irreducible factor (over Q) of x^n + (x + 2)^n.
Suppose f(alpha) = 0. If beta = (alpha + 2)/alpha,
then degree(Q(beta)) <= degree(Q(alpha)).
Also beta^n + 1 = 0. Take it from there.

Peter:

Thanks. That's very neat! But it took me awhile to figure out that
x^n+1 is irreducible over Q when n > 0 is a power of two---a fact
needed to complete your proof.

Also, your argument applies if 2 is replaced by any nonzero rational
number.

--Edwin

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