TITLE: Origins of functional field arithmetic
QUESTION [9 upvotes]: Background: By function field, we mean a finite extension of the field of rational functions of one variable over a finite field with $p$ elements. Classfield theory for function fields was established by Chevalley in an Annals paper. An axiomatic characterization for number fields and function fields was established by Artin and Whaples, thus finally putting on firm ground the analogy between function fields and number fields. I have seen allusions that the germ of the idea was coming from Gauss. However since fields were not defined then, this was not a definitive statement.
Question: When was a definitive conjecture first made in mathematical history that there is a major analogy between algebraic number fields and function fields over finite fields?

REPLY [7 votes]: One of the most early observations on the analogy between function fields and number fields is the little known work by 

E. Heine, 
 Fernere Untersuchungen über ganze Functionen,
J. Reine Angew. Math. 48 (1854), 243--266 

who started investigating quadratic forms with coefficients in polynomial rings. 
Dedekind's and Weber's contributions have already been mentioned, but not those of Robert König, who investigated the connection between quadratic function fields and quadratic forms with polynomial coefficients in various publications prior to Artin
(and perhaps just as unknown as Heine's contribution):

 Über die quadratischen Formen mit rationalen Funktionen als Koeffizienten, 
Monatsh. f. Math. Phys. 23 (1912), 321-346
Beiträge zur Arithmetik der hyperelliptischen Funktionenkörper,
J. Reine Angew. Math. 142 (1913), 191-210

He also explicitly stated the analogy between function fields and number fields in the titles (and the body) of the following two papers

Arithmetisch-funktionentheoretische Parallelen,
Jahresber. DMV 23 (1914), 181--192
Funktionen- und zahlentheoretische Analogien,
Jahresber. DMV 28 (1920), 208-213.

The constant fields investigated by König have characteristic 0, however.