Q:

Polynomials $X^n$ and $(1-X)^n$ are coprime

Let $n\in \mathbb N-\{0\}$ . How can we show that the polynomials $X^n$ and $(1-X)^n$ are coprime? Do we have to do an induction on $n$? it is clear that $X$ and $(1-X)$ are coprime by Euclid's algorithm. We can write $(1-X)^{n+1}=\sum_{k=0}^{n+1}{(-1)^k{k\choose n+1}X^k}$ but then how to proceed for the induction. thank you for your help.

A:

The only irreducible divisor of $x^n$ is $x$. By the binomial theorem,
$$(1-x)^n=1+x\left(\sum_{k=1}^n(-1)^k{n\choose k}x^{k-1}\right).$$
But then if $x|(1-x)^n$ we have that $$x\bigg|\left((1-x)^n-x\left(\sum_{k=1}^n(-1)^k{n\choose k}x^{k-1}\right)\right).$$
Of course this means $x|1$ which is impossible. Hence no primes are shared by $x^n$ and $(1-x)^n$ so they are coprime.

A:

Let $Y = 1 - X$, we have
$$1 = (X + Y)^{2n-1} = 
\left( \sum_{k=0}^{n-1} + \sum_{k=n}^{2n-1} \right) \binom{2n-1}{k} X^k Y^{2n-1-k}$$
Let $j = 2n-1-k$ in the second sum and notice $\displaystyle\;\binom{n}{k} = \binom{n}{j}$, we get
$$1 = \left( \sum_{k=0}^{n-1} \binom{2n-1}{k} X^k Y^{n-1-k} \right) Y^n
+ X^n \left( \sum_{j=0}^{n-1} \binom{2n-1}{j} X^{n-1-j} Y^j\right) 
$$
Since $1$ can be expressed as a linear combination of $X^n$ and $Y^n = (1-X)^n$ over the
polynomial ring of $X$, $\gcd(X^n, (1-X)^n) = 1$.

A:

We have according to Newton's identity:
$(1-X)^n=\sum\limits_{k=0}^n{n\choose k}(-1)^kX^{n-k}=1+X\sum\limits_{k=1}^n{n\choose k}(-1)^kX^{n-k-1}$
Put $P=\sum\limits_{k=1}^n{n\choose k}(-1)^kX^{n-k-1}$
It means that $\exists P\in\mathbb{C}[X],\,(1-X)^n-P(X)X=1$.
According to Bezout's theorem, $\gcd\left((1-X)^n,X\right)=1$ and so $\gcd\left((1-X)^n,X^n\right)=1$ (because if $\gcd(P,Q)=1$ then $\forall(n,m)\in\mathbb{N}^2,\,\gcd(P^n,Q^m)=1$).

