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There are no pages for quadratic polynomial and cubic polynomial because they are redirects. There are, however, pages for quadratic function and cubic function and there are pages for quadratic equation and cubic equation. Regardless of what the "correct" term is, if you actually read the content of those articles, it is very clear that one set is more relevant to a discussion about the discriminant of a polynomial than the other. I tried making the very obvious link correction along with a very detailed explanation but it was swiftly reverted by someone who left a note indicating that they had not done their due diligence.

Might seem like a small thing but I'm actually studying this topic at the moment. And quadratic function and cubic function were unhelpful enough for me track down the correct pages so I figured I'd save the next person some time. Robo042 (talk) 00:57, 13 July 2021 (UTC)Reply

Which page do yo need to track for learning about discriminants? The page Discriminant is there for that. If some information is lacking, then it must be added to this page. However Wikipedia is not a textbook; so for learning a subject, it is often better to work on a textbook.

Apparently, you have not understood that the wikilinks in the text of the article are not here to signal related articles, but for providing a definition of the technical terms that are used (here quadratic polynomial and cubic polynomial). This is a big difference between Wikipedia and a textbook, because, in a textbook, the author knows which terms are supposed to be known by the reader, and must define the others. Here, it is true that the theory of discriminants and the theory of polynomial equations are strongly related, and that, in degree 2 and 3, all properties of a polynomial equation result from its discriminant. Therefore you are right that quadratic equation and cubic equation must be linked. It is the reason for which I have linked these pages, and more specifically their sections "Discriminant" at the top of the respective sections of this article, by mean of a {{see also}} template. D.Lazard (talk) 12:21, 13 July 2021 (UTC)Reply

I'm reading to supplement my math studies. Something I would highly recommend to folks looking to improve their English skills as well.

I'm fine with your solution. And I would have been fine with your reverts if it wasn't very obvious that you had performed them before fully reading the detailed edit descriptions. In the future, please be sure to do your due diligence before reverting logical and well-explained edits.

Robo042 (talk) 12:38, 13 July 2021 (UTC)Reply

In this section it is stated that "Over the reals, if the discriminant is positive, then the surface either has no real point or has everywhere a negative Gaussian curvature. If the discriminant is negative, the surface has real points, and has a negative Gaussian curvature".

However, the correct statement shouldn't be "Over the reals, if the discriminant is positive, then the surface either has no real point or has everywhere a positive Gaussian curvature. If the discriminant is negative, the surface has real points, and has a negative Gaussian curvature"? For instance, taking A as the identity matrix we have the equation of a sphere, whose Gaussian curvature is positive everywhere. Saung Tadashi (talk) 14:11, 10 March 2023 (UTC)Reply

The formulation is correct, although it should be made clearer that the discriminant refers to the equation of the quadric in the projective space (homogeneous equation), and the Gaussian curvature refers to a non-homogeneous equation in 3D space. In the case of a unit sphere, the homogeneous equation is ${\displaystyle x^{2}+y^{2}+z^{2}-t^{2}=0,}$ and the discriminant is −1, as the determinant is of a 4×4 diagonal matrix that contains three 1 and one −1. D.Lazard (talk) 14:44, 10 March 2023 (UTC)Reply

If a polynomial with integer coefficients has a discriminant of D with |D| > 1 then, if the the polynomial is interpreted as within the field Z_p (i.e. within the field of integers modulo p), for some prime p that is a divisor of |D|, it will have a discriminant of 0 in that field. Thus it will have a root with multiplicity greater than 1 in that field, etc. Is this exploited in any way and, if so, would you point me towards a reference and/or add something about it to the current article? —Quantling (talk | contribs) 21:17, 7 May 2025 (UTC)Reply

"Thus it will have a root with multiplicity greater than 1 in that field": no, it will have a root with multiplicity greater than 1 in some field extension of ⁠${\displaystyle \mathbb {Z} _{p}}$⁠. The only application that I know of the situation that you describe is in Factorization of polynomials § Factoring univariate polynomials over the integers: for such a factorization, one must choose a prime; all primes but a finite number are good choices, since the bad choices are the prime divisors of the discriminant and the leading coefficient. Note that the discriminant need not to be computed, because the algorithm would detect automatically a bad choice. D.Lazard (talk) 09:12, 8 May 2025 (UTC)Reply

Thank you for the explanation! —Quantling (talk | contribs) 14:49, 8 May 2025 (UTC)Reply

I can imagine a connection, so I will ask: I see in Fibonacci sequence § Divisibility properties that whether a prime p evenly divides F_p−1, F_p, or F_p+1 depends upon the value of p mod 5. Is this related to the fact that the discriminant for the associated polynomial 1 − x − x² is 5? If yes, and if the explanation is easy enough (as well as well cited and/or WP:CALC, rather than WP:OR), I'd like to add that information to that article. —Quantling (talk | contribs) 17:17, 16 May 2025 (UTC)Reply

"to distinguish it from the of an extension K/L of number fields. "

from the what? Jesuslop (talk) 20:45, 1 August 2026 (UTC)Reply

Fixed. D.Lazard (talk) 20:55, 1 August 2026 (UTC)Reply