Mustafa Eren Taşlı, Dokuz Eylül University. Date: 26th of December, 2024, Wednesday, Time: 15.00 – 16.00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Classroom B255.
Abstract: We will define symmetric polynomials and the elementary symmetric polynomials in 
indeterminates over a field 
. The elementary symmetric polynomials in the indeterminates 
are as follows:
The Fundamental Theorem of Symmetric Polynomials states that any symmetric polynomial can be expressed as a polynomial in the elementary symmetric polynomials, that is:
Theorem. Let 
be a symmetric polynomial in the indeterminates over a field . Then, there exists a polynomial 
in the indeterminates 
such that
![\[f(x_1, x_2, \ldots, x_n) = g(\sigma_1, \sigma_2, \ldots, \sigma_n),\]](https://quicklatex.com/cache3/86/ql_7204178dea0bfa639531ef8835e31786_l3.svg "Rendered by QuickLaTeX.com")
where 
are the above elementary symmetric polynomials of the indeterminates . Moreover, the polynomial is uniquely determined.
We will prove this theorem using the graded lexicographic order for multivariable polynomials.
Using the recurrence relation from the Newton Identities, we will learn how to express the sum of powers of the indeterminates, that is, the polyomials
![\[s_k = x_1^k + x_2^k + \ldots + x_n^k\]](https://quicklatex.com/cache3/1c/ql_b3947574d56896a407d99b12cb82c51c_l3.svg "Rendered by QuickLaTeX.com")
for a positive integer 
, as polynomials in terms of the elementary symmetric polynomials. We will reinforce this understanding with examples.
The discriminant in the indeterminates over the field is given by:
![\[\Delta = \prod_{1 \leq i < j \leq n} (x_i - x_j)^2 \in F[x_1, \dots, x_n].\]](https://quicklatex.com/cache3/43/ql_682efe13d812226573dc8652daa38243_l3.svg "Rendered by QuickLaTeX.com")
The discriminant is a symmetric polynomial, and we will express it in terms of the elementary symmetric polynomials using determinants.
This seminar, as part of my graduation project titled Symmetric Polynomials, Newton’s Identities, Discriminants, and Resultants, serves as an introduction to a method for calculating the discriminant (
) of an -th degree polynomial without finding its roots.
![\[ x^3 + bx^2 + cx + d = 0, \]](https://quicklatex.com/cache3/6a/ql_2e2f8cd84c4ce76efb7fb9b06206566a_l3.svg "Rendered by QuickLaTeX.com")
![\[ y^3 + py + q = 0, \]](https://quicklatex.com/cache3/bf/ql_0d8db80131ada767debc90a3981a2fbf_l3.svg "Rendered by QuickLaTeX.com")
![\[ y_1 = \sqrt[3]{\frac{-q + \sqrt{q^2 + \frac{4p^3}{27}}}{2}} + \sqrt[3]{\frac{-q - \sqrt{q^2 + \frac{4p^3}{27}}}{2}}, \]](https://quicklatex.com/cache3/46/ql_b0a5d9db041094f1b16bdb2e8028ef46_l3.svg "Rendered by QuickLaTeX.com")
![\[ y_2 = \omega \sqrt[3]{\frac{-q + \sqrt{q^2 + \frac{4p^3}{27}}}{2}} + \omega^2 \sqrt[3]{\frac{-q - \sqrt{q^2 + \frac{4p^3}{27}}}{2}}, \]](https://quicklatex.com/cache3/71/ql_a0584521c6ef0017ae10d8afff37fe71_l3.svg "Rendered by QuickLaTeX.com")
![\[ y_3 = \omega^2 \sqrt[3]{\frac{-q + \sqrt{q^2 + \frac{4p^3}{27}}}{2}} + \omega \sqrt[3]{\frac{-q - \sqrt{q^2 + \frac{4p^3}{27}}}{2}}, \]](https://quicklatex.com/cache3/97/ql_2bb3ced4987531f3ba5c2ff53bea4d97_l3.svg "Rendered by QuickLaTeX.com")
is a primitive cube root of unity and where the above cube roots are one of the three complex cube roots whose product is 
and these are fixed in the above formulas.
, the discriminant is expressed by:![\[ \Delta = -27q^2 - 4p^3 = (y_1-y_2)^2(y_1-y_3)^2(y_2-y_3)^2. \]](https://quicklatex.com/cache3/46/ql_d93829faf28fbe23838ebf1bbf44b746_l3.svg "Rendered by QuickLaTeX.com")
, whose roots are 
, the discriminant is expressed by:![\[ \Delta = b^2c^2 + 18bcd - 4c^3 - 4b^3d - 27d^2 = (x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2. \]](https://quicklatex.com/cache3/04/ql_e7e22903bb3446549f52a74c9bcd9604_l3.svg "Rendered by QuickLaTeX.com")
(or 
) are in a subfield of 
, the cubic polynomial is irreducible over 
; and a minimal generating set of M is any generating set Y of M such that no proper subset of Y can generate M.