non-zero regulator

This is a question on page 14 of Thaine's article on Ideal class groups of real abelian number fields in Annals of Math, 128.

Leave $$Delta = { sigma_0 = id, sigma_1, .., sigma_r }$$, where $$Delta = Gal (K / mathbb {Q})$$Y $$K$$ It is a real Abelian numeric field.

Consider the polynomial $$f (X_1, .., X_r) = det mid X_ {p (i, j)} mid_ {1 leq i, j leq r}$$where the integers $$p (i, j), 0 leq p (i, j) leq r$$ are defined by $$sigma_ {i, j} = sigma_ {p (i, j)}$$ Y $$X_0 = – X_1 – … – X_r$$.

The paper claims. $$f (1, .., 1) = pm mid Delta mid ^ { Delta -2}$$.

I do not understand the claim.