Intereting Posts

The Average Height of an American is (Hypothesis Test)
Last Digits of a Tetration
Square Fibonacci numbers
Given a finite Group G, with A, B subgroups prove the order of AB
Fibonacci Sequence problem. Prove that there are infinitely many prime numbers such that $p$ divides $F_{p-1}$
A finite Monoid $M$ is a group if and only if it has only one idempotent element
In Rudin's proof of the completeness of $L^\infty$
Description of the kernel of the tensor product of two linear maps
Riemann zeta function at odd positive integers
Given K balls and N buckets what is the expected number of occupied buckets
Axiomatic characterization of the rational numbers
What is the transformation representation/interpretation of symmetric matrices?
Non-Circular Proof of $\lim_{x \to 0} \frac{\sin x}{x} = 1$
Proving $\pi^3 \gt 31$
What is the integral of 0?

I would like to if the polynomials of the form $x^{2^{n}}+1$ are irreducible over $\mathbb{Q}$ and in that case if there is some “easy” proof for that (where easy means not using a big theory like Galois).

- Decomposition of polynomial into irreducible polynomials
- Find the number of irreducible polynomials in any given degree
- How many irreducible monic quadratic polynomials are there in $\mathbb{F}_p$?
- How to solve Diophantine equations of the form $Axy + Bx + Cy + D = N$?
- Why this polynomial is irreducible?
- Simple extension of $\mathbb{Q} (\sqrt{2},i)$
- Irreducibility of a polynomial if it has no root (Capelli)
- Showing $(a+b+c)(x+y+z)=ax+by+cz$ given other facts
- Prove that $f(x)$ is irreducible iff its reciprocal polynomial $f^*(x)$ is irreducible.
- Show $x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $\mathbb Q$.

Let $f(x)=x^{2^n}+1$. Note that if $f(x+1)$ is irreducible, then so is $f(x)$.

We have:

$$f(x+1)=\left(x^{2^n} + \binom{2^n}{1}x^{2^n-1}+\dots+\binom{2^n}{2^n-1}x+1\right)+1$$

Note that $2$ divides all the coefficients except that of $x^{2^n}$, and $4$ does not divide the constant coefficient, $2$. Thus the polynomial is irreducible by Eisenstein’s criterion.

- Finding the 2,147,483,647th prime number
- Upper bounds for the number of intermediate subgroups
- Does the intersection of two finite index subgroups have finite index?
- Integrate $e^{-\frac{y^2}{2}}\left(\frac{1}{y^2}+1\right)$
- Cauchy-Formula for Repeated Lebesgue-Integration
- Evaluate $\int_0^{\frac{\pi}{2}}\frac{x^2}{1+\cos^2 x}dx$
- Finding the (unit) direction vector given azimuth and elevation
- An explicit construction for an everywhere discontinuous real function with $F((a+b)/2)\leq(F(a)+F(b))/2$?
- Sequence of monotone functions converging to a continuous limit, is the convergence uniform?
- Fredholm operator norm
- Find out the angle of <ABC
- composition of certain covering maps
- Heuristics for topological sort
- convergence/or divergence of the series $\sum_{n=1}^{\infty}(a_n+b_n)$ given the convergence/divergence of component series
- How do you determine if a point sits inside a polygon?