Talk:Discriminant

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[edit] Discriminant of a polynomial

I have found that the TeX code does not compile if the matrix has more than 10 colons. Maybe there is a way around this? The following matrix seems, to me, sufficient for comprehension:

\begin{pmatrix}   1     & a_{n-1}          &  a_{n-2}        &   .          &   a_{0}            &    0                &   .               &  .  &  .       &  0 \\   0     & 1                   &  a_{n-1}        &   .          &  .                   &   a_{0}            &   0              &  .  &  .       &  0 \\   0     & 0                   &  1                 &   .          &  .                   &   .                  &   a_{0}         &  0 &  .       &  0 \\   .      & .                    &  .                  &   .          &  .                   &   .                  &   .                & .   &  .       &  .  \\   .      & .                    &  .                  &   .          &  .                   &   .                  &   .                & .   &  .       &  .  \\   0     & 0                   & . . .               &  1          &  a_{n-1}         & a_{n-2}          &   .                & .   &  .       & a_{0} \\   n     & (n-1) a_{n-1} & (n-2) a_{n-2}&  . . .       &  0                  & 0                   &   .                & .   &  .       & 0 \\   0     &  n                  & (n-1)a_{n-1} &  . . .       &  1 a_1            & 0                   &   0               & .   &  .       & 0 \\   .      & .                    &  .                  &   .          &  .                   &   .                  &   .                & .   &  .       &  .  \\   .      & .                    &  .                  &   .          &  .                   &   .                  &   .                & .   &  .       &  .  \\   0     & 0                   &  . . .              & n           & (n-1)a_{n-1}  &  (n-2)a_{n-2} &   .                & .  & 1 a_1 & 0 \\   0     & 0                   &  . . .              &  0          &  n                  &  (n-1)a_{n-1} & (n-2)a_{n-2}& .  & .        & 1 a_1  \\ \end{pmatrix}

and note that there should be an (n-2) factor to the a_{n-2} terms in the two last rows. If there is no opposition, I think that this should replace the current "array". Gene.arboit 03:17, 12 August 2005 (UTC)

I combined some dots into \ldots, and compressed it into less than 10 columns, see if the matrix is comprehensible. Wang ty87916 15:55, 2 September 2006 (UTC)

[edit] discriminant of an algebraic number field

The stuff on the discriminant of an algebraic nubmer field should be a separate article. Although there are connections between discriminants of polynomials and of number fields, they are really two quite separate topics. Dmharvey Image:User_dmharvey_sig.png Talk 12:54, 29 August 2005 (UTC)

There may be a reason to split that out; but the 'separateness' is quite debatable. Charles Matthews 14:25, 29 August 2005 (UTC)

[edit] Latex Help

Help! My LaTex isn't working! I'm trying to LaTex the major formulas,but it says: Failed to Parse on some of them.

It appears that you somehow inserted some invisible unicode characters. Maybe you accidentally hit some weird key combination. In any case, I have reverted the changes, according to the guidelines at Wikipedia:Manual_of_Style_(mathematics). Dmharvey 08:08, 1 January 2006 (UTC)


[edit] Extra Topic Needed

Anyone know about Discriminant Functions in Statistics?

Please add, if you do.

[edit] Disambiguation

As there are many types of discriminants in mathematics: polynomial discriminant, elliptic discriminant, modular discriminant, fundamental discriminant, conic section discriminant, metric discriminant, etc. see MathWorld. So can we move the main portion of this article to Discriminant (polynomial) and create separate articles for all other discriminants, and set up an disambigution page?

[edit] Confusing article tag

I have to agree that this article is very confusing as it stands. If some high-school freshman wanders in here trying to read about the discriminant his algebra teacher told him about, the 10x10 matrix will probably give him a case of the fantods, and drive him away from dry-as-dust mathematics for the rest of his life!

I think the headline article discriminant should present a simple explanation for maybe the quadratic and cubic polynomials, and move the results from complex analysis / higher theory of polynomial equations into a separate section at least, or maybe even a separate article. I'll give it a go, but thought I'd ask for feedback first. ;^> DavidCBryant 14:18, 13 December 2006 (UTC)

Well, I went ahead and rewrote the article. It can still use lots of improvement, I'm sure. I added a section having some formulas (the ones that one always needs to look up). I put the quadratic formula stuff in a subsection. Now the general definition is further down, and that may make the article less offputting. I also fixed the definition (I think). Previous versions claimed the discriminant is equal to the resultant of p and p'. That is not true. --345Kai 10:17, 14 December 2006 (UTC)

[edit] Error on the first sentence?

I didn't want to edit without checking with the community first, but in the first sentence it says the discriminant is zero if and only if the polynomial *does* have multiple roots in the complex numbers, which is the opposite of the right answer, right? Omgoleus 14:23, 8 February 2007 (UTC)

[edit] The whole paragraph is wrong

The following paragraph is not easy to fix, but does not belong to the lead anyway, in my opinion:

The concept of discriminant has been generalized to other algebraic structures besides polynomials, including conic sections, quadratic forms, and algebraic number fields. Discriminants in algebraic number theory are closely related, and contain information about ramification. In fact, the more geometric types of ramification are also related to more abstract types of discriminant, making this a central algebraic idea in many applications.
  • Conic sections are not algebraic structures, while their discriminants are discriminants of polynomials.
  • Quadratic forms are polynomials. However, they are polynomials of several variables. Is this what caused the confusion?
  • Discriminants of algebraic number fields are, of course, part of algebraic number theory.
  • The sentence about ramification is extremely obscure.

I propose to delete the whole paragraph. Arcfrk 16:20, 22 March 2007 (UTC)

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