Understanding a sentence in a Galois Theory paper about polynomials

Question

I've been trying to understand a sentence of a math for some time. I'll be straightforward:

(*) $$x^n + a_{n-1}x^{n-1} +...+a_1x + a_0 = 0$$

(...)

If we denote the roots of (*) by $x_1, x_2, ... , x_n$ so that

$$(x-x_1)(x-x_2)...(x-x_n) = x^n + a_{n-1}x^{n-1} +...+a_1x + a_0$$

---

Then $a_0, ... , a_{n-1}$ are polynomial functions of $x_1, x_2, ... , x_n$ called elementary symmetric functions:

$$a_0 = (-1)^nx_1x_2...x_n , a_{n-1} = -(x_1 + x_2 + ... + x_n)$$

---

This last statement between the two lines is the one I have some issues with.

1. When the author claims that $a_0, ... , a_n$ are functions of $x_1, ... , x_n$, does he simply mean it in a way that $x_1, ... , x_n$ define what the elements $a_0, ... , a_n$ are?

2. And more importantly, where does this last equation come from? The only way I can think of making a_0 a function of $x_1, ... , x_n$ and $a_1, ... , a_n$ is to use (*) to write

$$a_0 = (-1)[x^n + a_{n-1}x^{n-1} +...+a_1x]$$

, but I can't seem to relate this last equality to the one the author wrote.

---

I would truly appreciate any help/thoughts!

Answer 1

I assume that the author means: Given the roots $x_1,...,x_n$, I can calculate the $a_0,...,a_{n-1}$. What you are basically doing is just multiplicating out the right hand side of the equation: $$ x^n + a_{n-1}x^{n-1} +...+a_1x + a_0=(x-x_1)(x-x_2)...(x-x_n) $$ Then the only term, which has degree $0$ on the right side is $\prod\limits_{i=1}^{n} (-x_i)=(-1)^n\prod\limits_{i=1}^{n}x_i$, thus it has to be equal to the term of degree $0$ on the left side, which is $a_0$. The only terms of degree $n-1$ on the right side are of the form $ x^{n-1}\cdot (-x_i)$. Summing them up you get $(-\sum\limits_{i=1}^{n}x_i)x^{n-1}$ and comparing to the term of degree $n-1$ on the left side, $a_{n-1} x^{n-1}$, you get $a_{n-1}=-\sum\limits_{i=1}^{n}x_i$

Answer 2

For $1$, what darij said.

Just expand $(x-x_1)(x-x_2)\cdots (x-x_n)$. For small $n$, we have:

$$(x-x_1)(x-x_2) = x^2 - (x_1 + x_2) x + x_1 x_2 \\(x-x_1)(x-x_2)(x-x_3) = x^3 - (x_1 + x_2 + x_3)x^2 + (x_1 x_2 + x_1 x_3 + x_2 x_3)x - x_1 x_2 x_3$$

and the pattern reveals itself as the author claims:

$$(x-x_1)\cdots(x-x_n) = x^n - (x_1 + \cdots + x_n) x^{n-1} + \cdots + (-1)^n x_1 \cdots x_n$$

now compare these two expressions of $x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$.