On the top of page 405 (413 of the pdf) of lecture notes by Professor Elman https://www.math.ucla.edu/~rse/algebra_book.pdf, it was asserted that $$\sum_{j=0}^m a_je^jF(0)-\sum_{j=0}^m a_jF(j)=-\sum_{j=0}^m a_jF(j).$$
I couldn't figure out why the first sum is zero. The function $F$ is defined as follows. Suppose $e$ is algebraic and let $m$ be the degree of an irreducible polynomial with integer coefficients that has $e$ as a root. Define $$f(x)=\frac{x^p(x-1)^p\cdots (x-m)^p}{(p-1)!}$$ for some prime number $p>2$ and $$F(x)=f(x)+f'(x)+\cdots +f^{(mp+p-1)}(x).$$
This is because the numbers $a_j$ were chosen so that $a_me^m + a_{m-1}e^{m-1} + \cdots + a_1e + a_0 = 0$.