# On coefficients of $(t-x_1)(t-x_2)..(t-x_i)$

I was reading Emil Artin's Galois Theory (2nd edition). On pp.39-40, Artin defines a number of things as follows.

Let $$k$$ be a field and $$E=k(x_1,\ldots,x_n)$$ be the field of all rational functions of $$n$$ indeterminates $$x_1,\ldots,x_n$$. Define a polynomial $$f(t)=(t-x_1)\cdots(t-x_n)=t^n+a_1t^{n-1}+\cdots+a_n,\tag{3}$$ where $$a_1=-\sum_ix_i,\,a_2=\sum_{i,... are elementary symmetric functions in $$x_1,\ldots,x_n$$. Then he puts $$F_n(t)=f(t)$$ and also for $$i=n-1,\,n-2,\ldots,1$$, $$F_i(t)=\frac{f(t)}{(t-x_{i+1})(t-x_{i+2})\cdots(t-x_n)}=\frac{F_{i+1}(t)}{t-x_{i+1}}.\tag{5}$$ (If I am not mistaken, this simply means that $$F_i(t)=(t-x_1)(t-x_2)..(t-x_i)$$ for each $$i$$, but what matters below is the division on the RHS of $$(5)$$.) He then writes that (emphasis mine):

Performing the division [in $$(5)$$] we see that $$F_i(t)$$ is a polynomial in $$t$$ of degree $$i$$ whose highest coefficient is $$1$$ and whose coefficients are polynomials in the variables $$a_1,a_2,\ldots,a_i$$ and $$x_{i+1},x_{i+2},\ldots,x_n$$. Only integer enter as coefficients in these expressions.

Can someone explain the bold part? I have thought about using Euclidean algorithm, but the $$a_i$$s are polynomials in $$x_1,\ldots,x_n$$ rather than constants. I am not sure how the division is done.

You just perform polynomial division as if the symbols were numbers. For example, you can check that $$(t^3+a_1t^2+a_2t+a_3)\div(t-x_3)=t^2+(a_1+x_3)t+[(a_1+x_3)x_3+a_2]$$ and $$\left(t^2+(a_1+x_3)t+[(a_1+x_3)x_3+a_2]\right)\div(t-x_2)=t+(a_1+x_2+x_3).$$