# Thread: Question on the History of a Notation

1. ## Question on the History of a Notation

Often when one does field theory, or at least this was my experience, one's first encounter notationally with a field is that they are generically denoted $\displaystyle F$ or $\displaystyle \mathbb{F}$. This of course makes sense for the same reason that vector spaces are generically denoted $\displaystyle V$, modules and monoids $\displaystyle M$, rings $\displaystyle R$, and groups $\displaystyle G$. Moreover, it is common (in accordance with sets) that these structures should have a capitalized letter. That said, it seems to be a universal notation among more advanced field theory books (especially those dealing in algebraic number theory) to denote fields by $\displaystyle k$--not f and not capitalized. Is there any particular reason for this? Is it possible that it is for the same reason that $\displaystyle \mathbb{Z}$ is the integer (zhalen)?

2. Hello,

Isn't it k because of the characteristic of a field ? (just a thought, I have no idea if it's correct or not)

3. Originally Posted by Moo
Hello,

Isn't it k because of the characteristic of a field ? (just a thought, I have no idea if it's correct or not)
Hey Moo,

That's an interesting hypothesis. I agree that in some of the books which denote fields generically by $\displaystyle F$ they write $\displaystyle \text{char}(F)$ as $\displaystyle k$--so that's definitely a possibility.

4. Originally Posted by Drexel28
Often when one does field theory, or at least this was my experience, one's first encounter notationally with a field is that they are generically denoted $\displaystyle F$ or $\displaystyle \mathbb{F}$. This of course makes sense for the same reason that vector spaces are generically denoted $\displaystyle V$, modules and monoids $\displaystyle M$, rings $\displaystyle R$, and groups $\displaystyle G$. Moreover, it is common (in accordance with sets) that these structures should have a capitalized letter. That said, it seems to be a universal notation among more advanced field theory books (especially those dealing in algebraic number theory) to denote fields by $\displaystyle k$--not f and not capitalized. Is there any particular reason for this? Is it possible that it is for the same reason that $\displaystyle \mathbb{Z}$ is the integer (zhalen)?
I think the answer lies in German, as with Zahlen for integers. The German word for field (in the algebraic sense) is Körper. Interestingly, the corresponding French word is corps, which (like Körper) means "body" in everyday usage.

5. Originally Posted by Opalg
I think the answer lies in German, as with Zahlen for integers. The German word for field (in the algebraic sense) is Körper. Interestingly, the corresponding French word is corps, which (like Körper) means "body" in everyday usage.
Yep, that would be pretty conclusive, thanks [b]Opalg[b/]! Any idea why the k is lower case though?

6. Just off the top of my head- Germans use lower case for symbols so as not to overuse capitals any more than they already do (in German all nouns are capitalized)!

7. I'm too lazy to do the research, but with all these German connections, my bet is that Gauss had a hand in a least a few of these cases.

-Dan

8. Originally Posted by topsquark
I'm too lazy to do the research, but with all these German connections, my bet is that Gauss had a hand in a least a few of these cases.

-Dan
Surprisingly, it seems to be very much more recent than that. The only online reference to the history of this notation that I have been able to find is here, where it is suggested that the first systematic use of Z for the integers and Q for the rationals is in Bourbaki's Algèbre. The Bourbaki group started work in the 1930s, but the first volume in the algebra series was not published until 1947. Before then, there does not seem to have been any standardised notation for number sets.