We seek analogs of the laws governing the behavior of prime ideals in extensions of algebraic number fields. Figure 1.5 shows what happens in a quadratic extension of the rationals. Figure 15.1 shows a non-normal cubic extension of the rationals. See Stark [118] for more information on these examples.

The graph theory analog of the example in Figure 15.1 is found in Figure 14.4 and Example 14.8. Figure 15.4 gives examples of primes that split in various ways in the non-normal cubic intermediate field.

So now let us consider the graph theory analog. The field extension is replaced by a graph covering Y/X, with projection map π. Suppose that [D] is aprimein Y. Then π(D) is a closed backtrackless tailless path in X, but it may not be primitive. There will, however, be a prime [C] in X and an integer f such that π(D) = Cf. The integer f is independent of the choice of D in [D].

Definition 15.1 If [D] is a prime in a covering Y/X with projection map π and π(D) = Cf, where [C] is a prime of X, we will say that [D] is a prime ofYabove [C] or, more loosely, that D is a prime aboveC (written as D∣C); f = f (D, Y/X) is defined as the residual degree of D with respect to Y/X.