Proof that e is transcendental in Herstein's Topics in Algebra (1st ed)

Question

In the proof of Theorem 5.F. page 177.

page 176 page 177 From the constructed $F(x)$ from $f(x),$ how can we choose $$f(x)=\frac{1}{(p-1)!}x^{p-1}(1-x)^{p}(2-x)^{p}\cdots(n-x)^{p}$$ where $p>n$ and $p>c_{0}.$

Any help would be very appreciated.

Answer 1

I think you are confused as to what depends on what. In the construction described in the two linked pages, Herstein shows how to construct a polynomial $F$ from any polynomial $f$. He then specifies that he will choose $f$ to be $$f(x)=\frac{1}{(p-1)!}x^{p-1}(1-x)^{p}(2-x)^{p}\cdots(n-x)^{p},$$ for some large prime $p$ larger than $n$ and $c_0$. As there are infinitely many primes, there exists some prime $p$ larger than both $n$ and $c_0$, and thus it is possible to construct that $f$ --- and from that $f$, it is possible to construct $F$.