Talk:Field (mathematics)

message: The last paragraph under Classic Definition attempts to express the Field axioms in terms of groups and whatnot.  It fails to confront the fact that 0 does not have an inverse in the multiplicative abelian group (and is therefore not a real group). 50.251.241.179 (talk) 01:44, 12 August 2018 (UTC)
replies: text: No, the phrase  handles the job perfectly. Purgy (talk) 07:48, 12 August 2018 (UTC)
thread_title:  Classic definition vagueness 

message: Can we throw the Parenthesis in the right side of the phrase, becuase we can rely on the fact that multiplication has priority over addition. Or because this is a "pure" definition of a field , we first have to use the parenthesis in the distribution, and only then talk about a convention that in a field,  multiplication has priority over addition? 93.173.65.88 (talk) 09:30, 2 August 2022 (UTC)
thread_title:  Parenthesis in the distribution axiom 

message: This article states, "as fields of rational functions."  However, this reference from nLab states "Rational functions on a field do not form a field.  This comes from the fact that the reciprocal function is not a multiplicative inverse of the identity function f(x)= \frac{x}{x} \ne 1 , due to the fact that f(x) is undefined at 0, and thus has a different domain than the constant function 1, which is the multiplicative unit." The rational function link is actually to rational fractions.  I recommend changing "rational function" to "rational fraction."  Any comments or suggestions?
https://ncatlab.org/nlab/show/rational+function#:~:text=Properties-,Rational%20functions%20on%20a%20field%20do%20not%20form%20a%20field,which%20is%20the%20multiplicative%20unit. TMM53 (talk) 00:51, 25 October 2022 (UTC)
replies: text: Your source seems to be assuming that the multiplication is pointwise.  That's not necessarily the case.  If you follow the link to field of fractions#Examples, you'll see (after a little unpacking) that we're talking about formal quotients of polynomials, with the multiplication defined in terms of the product of the numerators by the product of the denominators, up to a certain equivalence relation.  If you evaluate the result at a point, it will agree with the pointwise product, provided all subterms are defined, but the definition per se is not pointwise, and is not refutable by the argument your link gives. --Trovatore (talk) 02:07, 25 October 2022 (UTC)
thread_title:  Field of rational functions? 

message: This whole article uses capital F to represent an arbitrary field, except for this one specific section, § Geometry: field of functions, which uses lower-case k instead (but without mentioning the difference). This seems unnecessarily confusing. Is there some reason it can't also use F or at least capital K? –jacobolus (t) 19:46, 27 November 2023 (UTC)
replies: text: None that I can see, so I have replaced k by F. JBW (talk) 22:37, 27 November 2023 (UTC)
thread_title:  Can field-of-functions section use F for its field? 

message: I have tried for 5 minutes to understand the meaning of the tag you recently placed on this article, but failed.  Following the link to MOS:MATHSPECIAL was not illuminating.  What's the problem supposed to be?  --JBL (talk) 20:47, 30 January 2024 (UTC)
replies: text: This article uses Unicode character  in  blocks. According to MOS:MATHSPECIAL, those should be converted to  markup using \setminus or \smallsetminus for this character. I was not sure which is best, which is why I left the conversion to a math expert. (Looking back now, I do see \setminus in  blocks already in the article, so maybe that's our answer.) The rest of the  blocks containing this character also need to be converted to avoid complaints from other editors who think (and I agree) nesting  blocks inside  is too messy and hard to untangle. In doing these conversions, I find Help:Displaying a formula and general LaTeX syntax guides helpful. -- Beland (talk) 23:14, 30 January 2024 (UTC)
replies: text:  Thanks.  There is no distinction of meaning between \setminus and \smallsetminus, it's just a matter of authorial preference how to write it (similar to the various forms of lowercase epsilon or phi).  Personally I like smallsetminus.  I believe I've got every copy of the unicode character, but perhaps you could double check that I didn't miss any.  --JBL (talk) 17:15, 1 February 2024 (UTC)

text:  Looks good; thanks for your help! -- Beland (talk) 19:18, 1 February 2024 (UTC)
thread_title:  What does the tag mean?