# Talk:Characteristic (algebra)

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Field: Algebra

## Simpler explanations

Excuse me but this article sounds like gobbledygook to me. How about some simpler explainations for school kids preferably with examples.

There is a link to ring (mathematics). You need to know what a ring is in algebra. Follow that link. Michael Hardy 01:12, 13 Dec 2004 (UTC)
In response to your 'goobblygook' comment, I've changed the article to include a more approachable definition of a ring's characteristic. It would take quite of bit of instruction for school children to understand what a ring characteristic is, but if you are a school child don't let that hold you back! Start with groups first and work your way up to rings. If you aren't a school child forgive me for patronizing you.

## Examples needed for Abstract Algebra!

Can you please provide some example, a case in which there is non zero characteristic?

I feel that it would always be better if examples are cited along with theory in Abstract Algebra.

The article gives several examples of rings and fields with non-zero characteristic. The simplest is Z/nZ, the ring of integers modulo n, which has characteristic n. AxelBoldt 15:42, 23 May 2006 (UTC)

## Trivial Ring

Is the characteristic of the trivial ring defined? If so, it would seem to pose a problem to the statement that a ring has the same characteristic of its subrings. Mickeyg13 00:54, 28 April 2007 (UTC)

The characteristic of the trivial ring should be 1. There is no problem as the trivial ring isn't a subring of anything (except itself). Recall that subrings must contain the multiplicative identity. -- Fropuff 02:47, 28 April 2007 (UTC)
Bah, well in my mind, rings need not have a multiplicative identity, thus the trivial ring is a subring of every ring. I know the Wikipedia convention on this, but I don't have to agree with this, and I was forgetting about this when I posted. I wish the mathematical community could come to some sort of consensus on this. Mickeyg13 05:28, 28 April 2007 (UTC)
Ah, well if you omit the unital axiom then the statement certainly wouldn't be true (even if you disallow the trivial ring). -- Fropuff 06:01, 28 April 2007 (UTC)

## Definition of Characteristic Question

If a field (e.g. GF(2)) has elements 0 and 1, its characteristic is 2, because, as correctly stated in the article, "That is, char(R) is the smallest positive number n such that ..." followed by an image defining n as the number of summands. The image shows 1+1 = 0, where it takes two 1s (multiplicative) to get 0 (additive), hence characteristic n=2. However, before this explanation, the definition "is defined to be the smallest number of times one must add the ring's multiplicative identity element (1) to itself to get the additive identity element (0)." I don't speak math very well, but, to me, this is suggesting the characteristic should be 1. For GF(2), how many times must 1 be added to 1 to get 0? Once. That's not 2. Am I misreading the explanation? Thanks. --Karl Leifeste (talk) 16:03, 21 May 2009 (UTC)

I believe the editor wasn't considering the fact that 1+1 was adding 1 to itself once, instead of twice. What is meant is the minimum number of ones in a sum of ones. It appears an editor has now corrected this [1]. The phrasing is currently a bit awkward, but at least is accurate/unambiguous now. Nice catch. If you can think of a batter way to phrase this, that would be great. Thanks. RobHar (talk) 20:53, 3 August 2009 (UTC)

## Mixed characteristic

There are several pages that link here for a definition of the term mixed characteristic, but there is none in the article itself. It would help to put this in, because I've not been able to find a definition anywhere (although I have suspicions about what it might mean). —Preceding unsigned comment added by 163.1.180.170 (talk) 15:58, 8 October 2010 (UTC)

I don't really think it would make sense to define the term mixed characteristic here. An example of the use of the term is for local rings. A local ring (R with maximal ideal m) is of mixed characteristic if the characteristic of R is 0 and the characteristic of R/m is non-zero. I think it would make sense to define mixed characteristic in the article on local rings, say, or perhaps in the glossary of scheme theory. Doing a google search I only found four (math) articles with the term "mixed characteristic" and only one instance of a link to this page, and the link was only for the term "characteristic". This instance is Rational singularity, what are the other pages you are talking about? RobHar (talk) 03:55, 9 October 2010 (UTC)

## Prime fields

It seems that there are different ways to define prime fields. From this page it seems that a prime field is a field that does not contain any proper subfields, i.e. it is either GF(p) or Q. But some other pages on wikipedia, as well as a page on wolphram.com, define prime fields as GF(p). So, the unclear thing is: is Q a prime field after all? — Preceding unsigned comment added by 95.143.213.113 (talk) 06:28, 11 March 2012 (UTC)