# What's the name for this method of putting the roots of a polynomial in a system of equations?

As I was (and still am lol) struggling on the exercises for symmetric polynomials in my abstract algebra book, I stumbled upon this neat relating the solutions of a polynomial in $$\mathbb C$$ with the (real) coefficients of the polynomial (I'm sure this can be generalized to complex coefficients). As an example, consider the polynomial $$f (x)= x^3 - 5x^2 + 4x- 3$$. The fundamental theorem of algebra says that there are 3 roots, which we denote $$u, v,$$ and $$w$$, so $$f(x) = (x-u)(x-v)(x-w)$$.

In the polynomial ring $$\mathbb R[t, x_1, x_2, x_3]$$, consider $$f$$ but instead of $$u,v,w$$ we have $$x_1,x_2,x_3$$. $$(t - x_1)(t - x_2) (t - x_3).$$

I don't know how to cite a book but there's this theorem (4.5.3) in Introduction to Abstract Algebra 4th edition by Nicholson that says

Write $$s_k = s_k(x_1, \dots, x_n)$$ for $$1 \leq k \leq n$$. Then $$(t - x_1) (t - x_2) \cdots (t - x_n) = t^n - s_1 t^{n-1} + s_2 t^{n-2} - \cdots \pm s_n = \sum_{k=0}^n (-1)^k s_k t^{n-k}.$$

Here, $$s_k$$ represents the $$k$$th symmetric elementary polynomial in $$\mathbb R[x_1, x_2, x_3]$$. In this case we have

$$(t - x_1) (t - x_2) (t - x_3) = t^3 - s_1 t^2 + s_2 t - s_3.$$

Remember that this is supposed to be $$f$$, so the coefficients must match. Hence we set up the following system of equations. $$s_1(x_1, x_2, x_3) = 5 \\ s_2(x_1, x_2, x_3) = 4 \\ s_3(x_1, x_2, x_3) = 3$$

If we consider $$f$$ in $$\mathbb R$$ again, then we should have $$x_1 = u, x_2 = v,$$ and $$x_3 = w$$. So, by expanding the $$s_i$$ as well, we get the system $$u+v+w = 5 \\ uv + vw + vw = 4 \\ uvw = 3$$

And of course this can be generalized to polynomials that are not monic or have an alternating sign, so basically all of them. By approximating the solutions using Wolfram Alpha (link 1 and link 2) (just as a sanity check), these are definitely the roots of the polynomial, expressed in a neat system involving only the roots and coefficients! Is this already a known method? I'm sure it is, so what's its name?

• They are the ring-theoretic (universal) form of Vieta's formulas - known in high-school - long before one studies symmetric polynomials or Galois theory. Commented Nov 2, 2023 at 3:30
• @BillDubuque Thanks! i just rediscovered them myself lol Commented Nov 2, 2023 at 3:31
• A far more striking example of the universal power of polynomials is here where we use matrices with indeterminate coefficients to prove Sylvester's determinant identity. Analytic bias here is so strong that this purely algebraic proof often confuses many at first glance - including even some grad students and professors who have studied abstract algebra - who wrongly think it involves division by zero. Commented Nov 2, 2023 at 3:58

You have rediscovered the ring theoretic generalization$$^\dagger$$ of the formulas that are usually attributed to 16th century French mathematician François Viète, although special cases of this—certainly for quadratic polynomials—were known to Persian and Arab mathematicians centuries earlier. They're usually called Vieta's formulas, using a Latinized form of his name.
$$^\dagger$$See comment by Bill Dubuque.
• They're not Vieta's (original) formula's but rather the universal ring theoretic generalization of such, i.e. the "roots" $\,x_i\,$ are not "numbers" but rather indeterminates. Commented Nov 2, 2023 at 3:34