Talk:Algebraic equation

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Mathematics rating: Stub Class High Priority  Field: Algebra

I'm not familiar with the usage of "algebraic equation" to mean an equation with rational coefficients. If anything, it should be "a polynomial equation (with coefficients in whatever the working base field is)". Do you have a source for the definition given? Dmharvey 02:27, 6 July 2006 (UTC)

MathWorld has a slightly different but equivalent definition. I also found several university homework questions that used the same definition. This page was previously a redirect, linked to by a score of other wikipedia pages. In the contexts that those pages use "algebraic equation" (e.g. in galois theory) they are referring to this definition rather than a generalised one. (I did add several of those references myself, but mostly I was just turning "algebraic equation" into a link.) And I've not seen "algebraic equation" used in the more general sense you describe; they are just "polynomial equations". So I'm fairly sure this definition is correct but I'm not a professional mathematician so I wouldn't stake my life on it. Reilly 14:35, 6 July 2006 (UTC)

  • I think the term you want is algebraic polynomial, but it depends on what field you are considering. To take the extremes: Polynomials with rational coefficients are algebraic over Q, but any polynomial is algebraic over C. Septentrionalis 20:35, 6 July 2006 (UTC)

If an algebraic polynomial can have coefficients from any field, is algebraic polynomial just a synonym for polynomial? "Polynomials with rational coefficients are algebraic over Q"; is that right? It's a very long time since I studied any of this :-) Reilly 01:29, 7 July 2006 (UTC)

  • No; being algebraic is a relation between a polynomial and a field; so it depends on what field you are talking about, whether a polynomial is algebraic or not. Septentrionalis 02:22, 7 July 2006 (UTC)

Isn't "algebraic over" a relation between two fields? i.e. F is algebraic over G if all members of F are solutions of polynomial equations with coefficients in G. And this is a fairly modern usage? AFAIK, the algebraic in "algebraic equation" originally meant "composed from algebraic operations" (i.e. plus, multiply, etc.). Reilly 02:52, 7 July 2006 (UTC)

Yes, algebraic is also a relation between fields: E is algebraic over F iff all elements of E are roots of algebraic polynomials over F. Hence the extension of meaning. Septentrionalis 13:16, 7 July 2006 (UTC)

Thanks for answering my questions, and I'm sorry to bang on about this, but I'm still not getting it. What does it mean for a polynomial P to be algebraic over a field F? Does it just mean that the coefficents of P come from F? Reilly 15:16, 7 July 2006 (UTC)

Yes. (Do see if the article is clear; this is one reason non-mathematicians are needed in editing mathematical articles.) Septentrionalis 17:27, 7 July 2006 (UTC)

Thanks. I think the article is clear enough. Reilly 15:17, 10 July 2006 (UTC)

The concept of an algebraic polynomial is redundant and I don't think I've heard it used. Algebraic in this context means polynomial (as opposed to a transcendental function such as sine, cosine, the Riemann zeta function, etc. which can be expressed as power series or some other infinite expression). This is where algebraic geometry differs from say differential geometry, in the former one restricts to spaces and functions that can be defined in terms of polynomials, whereas in the latter one allows arbitrary differentiable functions. One should define an algebraic equation (over a given field) to be P=0 where P is a polynomial with coefficients in that field. RobHar 23:09, 16 December 2006 (UTC)

I am surprised by the claim that this is redundant. The algebraic nature of the equation also implies that the coefficients of the equation lie in the given field, or equivalently, in the algebraic closure of the field. I revert generally. Rob Har has made several other tweaks, but they seem to me indifferent; if they are put back, I don't care either way. Septentrionalis PMAnderson 22:31, 19 December 2006 (UTC)
Thanks for commenting on this, it has made me look further into the subject. Upon searching google and mathscinet for "algebraic polynomial" it does indeed appear that the term algebraic polynomial is used. However it does not appear to be used in a context germane to this article. Further, the generality of the definition of this article appears to be never used. I found my way to this article from articles related to algebraic geometry and that's why I made the claim that the term was redundant since in algebraic geometry (at least in my opinion) it is (one generally fixes a base field in algebraic geometry, and if one is speaking about more than one field, one would generally say the polynomial was "rational" over whichever field one wants). From searching the web, the term algebraic polynomial seems to be exclusively used to describe polynomials whose coefficients are algebraic numbers over the rationals. They seem to be mainly discussed when talking about approximating functions by such algebraic polynomials (for example Hermite polynomials), as on would approximate functions by trigonometric polynomials such as Fourier polynomials or Chebyshev polynomials. For this reason, I would suggest removing the definition of algebraic polynomial from this article and creating an article on its own, since it occurs in a different field of math. Anyone have any opnions? RobHar 19:37, 22 December 2006 (UTC)
I will however modify the two last paragraphs: for example, not every equation is alegebraic (the extent to which this is true would be techinically hard to describe, for example " True = not False" is not algebraic over any field). —The preceding unsigned comment was added by RobHar (talk • contribs) 19:50, 22 December 2006 (UTC).

[edit] High Importance Article

So someone just made this article a high-importance article which means it definitely needs to be changed. As can be seen from above, I have certain problems with this article that still remain (no one gave their opinion). For example, right now the term algebraic polynomial over a field means exactly the same thing as a polynomial over a field (thus my comment on it being redundant). As mentioned above as well, the term algebraic polynomial is not germane to this article as it is seemingly only used in approximation theory where it means polynomial over the complex numbers as opposed to a trigonometric polynomial for example. Also, encyclopedia britannica has a different definition of algebraic equation, so maybe there needs to be some references for this article. RobHar 22:05, 13 May 2007 (UTC)

Ok, so I'm just going to rewrite this article then. This is why:
  1. "Algebraic polynomial over a field" as defined here means exactly "Polynomial over a field".
  2. "Algebraic equation over a field" as defined here means exactly "Polynomial equation over a field".
  3. Furthermore, of the references I have found, algebraic polynomial and algebraic equation are used to contrast from two other possible concepts: a) transcendental polynomial (for example, a trigonometric polynomials), b) differential equation.
  4. Britannica has a different definition of algebraic equation (that does not discuss fields of definition), namely they allow taking nth roots as well as ratios.
Thus not only do I believe this definition is incorrect and unused, but also unverifiable (though there was a popular math book that somewhat agreed). Thus, I intend on saying an algebraic equation can mean one of two things in subfields of numerical analysis and mathematical analysis a) a contrast to transcendental equations, b) a contrast to differential equations. If you disagree with this, please bring reliable references to the table. Thanks. RobHar (talk) 22:53, 15 December 2007 (UTC)