# Racah G. 's Group Theory and Spectroscopy PDF

By Racah G.

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Sample text

N R) = w. Proof We define a homomorphism 1 T: R8-+- ZjZ w by mapping an element of the group algebra on its first coefficient mod Z. In other words, if we let TIX = a(I). Note that T(8) 1 1 == In - 2 (mod Z), and therefore that T is surjective. It now suffices to prove that its kernel is R8 n R. But we have whence for odd b prime to m, and IX T(UbIX8) E R, we get == bT(1X8) (mod Z). 29 2. () also lies in R, thereby proving the lemma. We now assume that m = pn is a prime power. / = R() () R is called the Stickelberger ideal We want to determine the index Define for any character Xon Z(m)*.

If x E ZeN) then x/N can be viewed be the smallest real number as an element of Q/Z. For any E R/Z we let ~ 0 in the residue class of t mod Z. What we want is for each positive integer k a polynomial PIc with rational coefficients, leading coefficient 1, such that the functions t form a distribution on the projective system {Z/NZ}. Such polynomials will be given by the Bernoulli polynomials. Let the Bernoulli numbers B" be defined by the power series B 1. 34 F(t) = tOOt" = B"" e - 1 "=0 k. -t- 2: §2.

Character Sums we get l/IE(Q) = ex(CO(P»Jl(Cl(P»))lE:F'l = l/I(p)m/r. With a view towards (2), we conclude that (3) TI (1 -l/IE(Q)xmn(Q») = (1 Q\P = l/I(p)m/rxmn/ry TI = (1 - l/I(P)(xny 1 {mf'~ TI ,m=l (l - l/I(p)(eX)n). For this last step, we observe that the map gives a surjection of {lm -7- {lm/n and the inverse image of any element of {lm/r is a coset of {lr since r = (m, n). This makes the last step obvious. Substituting (3) in (2), we now find 1 + SE(XE, JlE)X m = = JI IJ TI (1 (1 - l/I(p\(ex)n(p» + Sex, Jl)eX) ~m~l This proves the theorem..