Talk:Characteristic (algebra)

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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.

In response to the edit: IMHO it is still not easily understandable for anybody else then Mathematicians. I have a MSc. in Electrical Engineering, and do not recognize anything described from the mathematics I was taught. So, with regards to the initial question: Please simplify the explanation. Think of Count von Count-level. Stuff that non-mathematicians can read.131.155.227.142 (talk) 12:21, 24 October 2017 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Justification for term characteristic zero

The justifications given for why characteristic zero is a good convention would apply equally to groups. But infinite order is used there. — Preceding unsigned comment added by 2A00:23C7:F486:A00:697E:261B:A972:1BA7 (talk) 03:32, 9 May 2020 (UTC)[reply]

No justification was given for the phrase "characteristic zero". I have just added such an explanation. This explanation does not apply to group. On the contary, the order of a group is infinite if and only if the group is an infinite set, which makes the term "infinite order" natural. In any case, the main justification of a terminology is that it is commonly used and accepted mathematicians. In some cases, this comes from historical habits; in other cases, a terminology is chosen because it makes the fundamentals theorem easier to state without considering exceptions. Here, we are faced with a mixture of both. D.Lazard (talk) 09:29, 9 May 2020 (UTC)[reply]

Ring Isomorphism vs Field Isomorphism?

The article says "in fact, any field of characteristic zero and cardinality at most continuum is (ring-)isomorphic to a subfield of complex numbers."

Any two rings which are isomorphic will be isomorphic as fields. Fields add no additional structure to rings. This is quite confusing unless there is something I'm missing here. — Preceding unsigned comment added by Electro blob (talk • contribs) 22:01, 14 October 2021 (UTC)[reply]

The parenthese "(ring-)" was here (I guess) for emphasizing that the isomorphism does not extend to other properties. I have fixed the mathematical accuracy of this paragraph. D.Lazard (talk) 22:54, 14 October 2021 (UTC)[reply]

Short description too long

Hello, I'm currently cleaning up the 6000+ short descriptions that are over 100 characters. Short descriptions should be around 40 characters long and describe the subject in a nutshell. The short description for this article is 113 characters long and needs shortening. Usually I can understand from the lead what the general topic is, but I haven't been able to with this article. Could someone please change the short description to something short and appropriate like "Mathematical formula" or "Algebraic equation" etc it shouldn't be a summary of the article or the lead, just a nutshell description of the subject. Thanks. - X201 (talk) 08:58, 8 February 2022 (UTC)[reply]

Done D.Lazard (talk) 09:59, 8 February 2022 (UTC)[reply]

Thanks - X201 (talk) 10:05, 8 February 2022 (UTC)[reply]

"Equivalent" characterization

@Michael Hardy, in Special:Diff/498082361, added what is now worded as The characteristic of a ring $R$ is $n$ precisely if the statement $ka = 0$ for all $a \in R$ implies $k$ is a multiple of $n$ . This is not equivalent, since $ka = 0$ for all $a \in R$ clearly always implies that k is a multiple of 1 (or any other divisor of n), and therefore every ring would have characteristic one. I assume this should be changed to The characteristic of a ring $R$ is $n$ precisely if the statement $ka = 0$ for all $a \in R$ is true if and only if $k$ is a multiple of $n$ ? 1234qwer 1234qwer 4 21:57, 1 December 2022 (UTC)[reply]