Talk:Legendre symbol

Legendre symbol usually not defined for p=2

The Legendre (a/p) symbol is usually not defined for p=2. Moreover, the extended definition given here is not consistent with the Kronecker symbol. E.g., the Kronecker symbol (a/2) is either 1,0,-1 depending on a. But the Kronecker symbols is an extension of the Legendre and Jacobi symbol. Anyone agrees to remove the case p=2 from the definition? 24.228.93.22 17:12, 7 September 2005 (UTC)

If you believe you are correct, change the page, be bold. Greg321 21:58, 7 September 2005 (UTC)

The case p=2 is now removed from the definition. None of the references I checked defines the Legendre symbol for p=2. Besides the inconsistency with the Kronecker symbol, also Euler's equation would not be true for p=2. 24.228.93.22 02:06, 14 September 2005 (UTC)

Speaking as a reader unfamiliar with the topic and most of the topics referenced, I found it strange that 2 was excluded without even the slightest hint why. I'm sure there's a good reason for it, and this discussion gives a hint, but can anyone add an explanation (for the casual mathematically-inclined-but-unfamiliar-with-this-stuff reader) of exactly how it is problematic or ambiguous? --24.161.61.118 (talk) 20:28, 21 January 2011 (UTC)

It does not behave in the expected way. Basically, the whole point of the Legendre symbol is to have a convenient notation for quadratic reciprocity and similar properties of quadratic residues. That is, we are dealing with polynomial (quadratic) equations modulo a prime, and in such situations various oddities arise when the modulus divides the discriminant of the equation; in this particular case, the discriminant of x² − a is 4a, hence we need p odd (and not dividing a).—Emil J. 13:27, 25 January 2011 (UTC)

See the discussion of this at Math.StackExchange. Essentially defining the symbol with p=2 as base does not help Legendre formulate his law of quadratic reciprocity. Hardmath (talk) 11:37, 10 November 2014 (UTC)

Revision

I've been editing this and related pages quadratic residue, jacobi symbol adding notes, etc.

Virginia-American (talk) 07:26, 2 March 2008 (UTC)

I wonder if we should add something about how the Legendre symbol is expressed in various markup systems (HTML, Unicode, LaTeX), esp. since the name Legendre symbol suggests that the topic is notational. I've seen these markup issues being added to various math-related articles. Hardmath (talk) 11:43, 10 November 2014 (UTC)

Mistake?

I think there is a mistake here. The formula supposedly from Gauss, expressed in terms of the variable zeta, doesn't produce a result of 0 when it should. —Preceding unsigned comment added by 209.67.107.10 (talk) 01:11, 3 September 2008 (UTC)

Kronecker symbol be a generalization of Legnedre symbol?

How can Kronecker symbol be a generalization of Legnedre symbol? Shinjikun (talk) 16:39, 12 November 2009 (UTC)

Sorry I get it. Shinjikun (talk) 16:40, 12 November 2009 (UTC)

Should mention when a is in Fq2

Another generalization is when a is in $F_{p^{k}}$. Jackzhp (talk) 08:06, 16 January 2018 (UTC)

Why are there two tables?

Same comment applies to Jacobi symbol James in dc (talk) 02:58, 7 December 2023 (UTC)