A construction for absolute values in polynomial rings
Saunders MacLane · Transactions of the American Mathematical Society · 1936
An absolute value or "Bewertung" of a ring is a function ||6|| which has some of the formal properties of the ordinary absolute value.More explicitly, for any 6 in the ring, ||6|| must be a real number with the * Similar "second-stage" values V, appear implicitly in the irreducibility investigations of Ore [7], Kürschák [6], and Relia [l0].1936] * This term, used by Relia [10], is similar to Ore's "congruent modulo a polygon" ( [8 ], p. 270) and Kürschák's "equipollence" ([6J, p. 185).f Here and subsequently the element 0 plays an exceptional role.X Krull [14], p. 171, and Hasse-Schmidt [12], p. 31.* Henceforth all polynomials considered are to have coefficients in K, unless otherwise noted, t This assumption, although unnecessary, will simplify the subsequent work.* The details here omitted are given in Rella's proof.* These conditions involve no loss of generality, but simplify subsequent proofs (see Theorem 6.7 and the end of §9).