# Show some polynomial satisfies Eisenstein's Criterion

Consider a polynomial $$f(X) = X^{(p-1)p^{n-1}} + X^{(p-2)p^{n-1}} + \cdots + X^{p^{n-1}} + 1$$

Now I need to show $$f(X+1)$$ safeties Eisenstein's criterion. My argument is that
$$f(X+1) = (X+1)^{(p-1)p^{n-1}} + (X+1)^{(p-2)p^{n-1}} + \cdots + (X+1)^{p^{n-1}} + 1$$

I consider term by term, obviously, the coefficient of the highest order term is $$1$$, so $$p$$ does not divide $$1$$, the coefficient of the constant term is $$p$$, so $$p^2$$ does not divide $$p$$, for the middle terms, I observe that $$p$$ can mostly divide them, but for the first terms of each term of $$f(X+1)$$, coefficients for them are all $$1$$ ($$p$$ does not divide $$1$$) and I cannot easily find other equal order terms with them to observe the total coefficient, how should I solve it?

• "but for the first terms of each term of $f(X+1)$, coefficients for them are all $1$ ($p$ does not divide $1$)", are you sure about that? Do you have some example of this? – Sil Apr 20 at 22:41
• look at the second term, bu using binomial theorem, the first term of it is x^(p^n - 2p^(n-1)). so the coefficient of it is 1. p does not divide it. – Jonny Apr 20 at 22:46
• I am not sure what you mean by second term and first term here, can you write down the full polynomial that satisfies this? For example for $p=3,n=2$ we have $f(x+1)=x^6+6x^5+15x^4+21x^3+18x^2+9x+3$, I don't see the thing you describe. – Sil Apr 20 at 22:47
• I mean, if I take p=3, the first term of (X+1)$^{(P-2)P^{n-1}}$ is x$^{3^n - 2*3^{n-1}}$ – Jonny Apr 20 at 22:54
• I see, well that is just a part of the polynomial, you need to collect all coefficients (possibly by Binomial theorem), then it won't be just $1$, and Eisenstein will be applicable. – Sil Apr 20 at 22:56

Here is something that could get you started (too long for a comment), write out the polynomial as:

\begin{align*} f(X+1) &= (X+1)^{(p-1)p^{n-1}} + (X+1)^{(p-2)p^{n-1}} + \cdots + (X+1)^{p^{n-1}} + 1\\ &= \sum_{k=0}^{p-1}(X+1)^{kp^{n-1}}\\ &= \sum_{k=0}^{p-1}\sum_{i=0}^{kp^{n-1}}\binom{kp^{n-1}}{i}X^i\\ &= \sum_{i=0}^{(p-1)p^{n-1}}\sum_{k=\lceil i/p^{n-1}\rceil}^{p-1}\binom{kp^{n-1}}{i}X^i\\ \end{align*} where we just swapped the indices by writing the conditions $$0\leq k \leq p-1$$, $$0 \leq i \leq kp^{n-1}$$, and rewrote into equivalent $$0\leq i \leq (p-1)p^{n-1}$$,$$i/p^{n-1}\leq k \leq p-1$$. So coefficient of $$X^i$$ for $$i>0$$ can be seen to be $$\sum_{k=\lceil i/p^{n-1}\rceil}^{p-1}\binom{kp^{n-1}}{i}$$. Now for Eisenstein you need to show divisibility of this sum by $$p$$ for $$i<(p-1)p^{n-1}$$.

However I have noticed that in some cases it does not seem to be straightforward to prove, for example for $$p=3$$, $$n=2$$ and coefficient at $$x^3$$ we get get sum $$1+20=21$$, which is divisible by $$3$$ but individual terms in sum are not. So something else is needed.

Showing that all lower coefficients are divisible by $$p$$ is the easy half of the problem.
Your original polynomial is $$\frac{X^{p^n}-1}{X^{p^{n-1}}-1}\,.$$ and after the substitution it becomes $$\frac{(X+1)^{p^n}-1}{(X+1)^{p^{n-1}}-1}\,.$$ Now look at this modulo $$p$$, i.e. as a polynomial over $$\Bbb F_p$$, the prime field. The numerator becomes $$X^{p^n}+1-1=X^{p^n}$$, and similarly the denominator becomes $$X^{p^{n-1}}$$, so that the quotient has only one term not divisible by $$p$$.
To see that the constant term is $$p$$, look at numerator and denominator in the second display: both have constant term zero, so that you have a polynomial of the form $$\frac{X^{p^n}+\cdots+p^nX}{X^{p^{n-1}}+\cdots+p^{n-1}X}\,,$$ and this polynomial clearly has constant term $$p$$.
(I’m guessing that you came across this as the cyclotomic polynomial $$\Phi_{p^n}(X)$$. )