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Math Forum » Discussions » sci.math.* » sci.math.independent

Topic: Is x^n + (x+2)^n irreducible over Q if n is a power of 2?
Replies: 10   Last Post: Sep 12, 2006 12:43 AM

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Jannick

Posts: 1,307
Registered: 12/6/04
Re: Is x^n + (x+2)^n irreducible over Q if n is a power of 2?
Posted: Sep 11, 2006 3:42 AM
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On 11.09.2006 00:10, Ignacio Larrosa Cañestro wrote:
> En el mensaje:45047346.1000503@web.de,
> Jannick Asmus <jannick.news@web.de> escribió:
>> On 10.09.2006 20:02, Edwin Clark wrote:
>>> 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)=x^n +(x+2)^n, then f(y-1)/2 is an Eisenstein polynomial
>> w.r.t. 2, hence irreducible over Q (cf.
>> http://en.wikipedia.org/wiki/Eisenstein%27s_criterion).
>>
>
> But the constant term in f(y - 1)/2 is 1 ... Then you can't apply Eisenstein
> criterion.

Oupps, that's right !!! Sorry, I was too tired and hasty last night.

Best wishes,
J.

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