# Why do we use lowercase $k$ for fields?

As a general rule in mathematical notation, structures and collections are given uppercase names, while elements often have lowercase names. However, it's not uncommon for a field to be called $k$ (for example in this question). I know that this stands for the German word for field, Körper, but why lowercase?

• Little $k$ for the small field, big $K$ for the big field (say, the function field of a variety over $k$)... – Zhen Lin Apr 2 '14 at 20:23
• just a guess, but one extra ingredient in German was the prevalence of Fraktur for quite a while, still used for Lie algebras. Also, I always use upp er case for fields., so i think you are talking about specialists in some mathematics areas but not others. – Will Jagy Apr 2 '14 at 20:24
• @WillJagy: yes, I deliberately said "not uncommon" rather than "usual". F and K seem to be at least as common as k. – Max Apr 2 '14 at 20:27
• Just checked, Lam is consistent in always using upper case in Introduction to Quadratic Forms over Fields, AMS_GSM 67 (2004 or 2005). He uses a bar or a tilde or a dot directly over the letter to indicate related fields/things. Right, ~ is tilde, but Tilda Swinton. Also, PSG 3, Chelsea 1. – Will Jagy Apr 2 '14 at 20:32
• @WillJagy, the second sentence of Chapter 1 of Hartshorne's Algebraic Geometry says, "We work over a fixed algebraically closed field $k$." Similarly, Section 2 of Chapter V in Weil's Basic Number Theory begins, "From now on, until the end of this Chapter, $k$ will denote an algebraic number-field." Lang also uses $k$ to denote a field in his book Algebra, but I can't locate where he first does so (the first appearance I could easily find is on page 162). – Barry Cipra Apr 2 '14 at 21:13