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.