By ANDRE WEIL

From the experiences: "L.R. Shafarevich confirmed me the 1st version […] and acknowledged that this booklet could be any longer the publication approximately type box conception. in reality it really is via a ways the main entire therapy of the most theorems of algebraic quantity thought, together with functionality fields over finite consistent fields, that seemed in ebook form." Zentralblatt MATH

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However we may apply the result to both O('n) and its maximal real subfield, and then compare. ), and let dn+, h:, R: denote the corresponding objects for O('nt. We first estimate dn. 18. log dn = ~(n) log n + o(~(n) log n). Proof. 7, we have logd n = ~(n)logn - ~(n) logp L --. pin P - 1 Let m = log n/log 2. Since 2m = n, it follows that n has at most m prime factors. Clearly log P log Pi log Pi L ~~ L ~ ~ 2 L ---~ 1m] pin P - 1 i=l 1m] Pi - 1 i=l Pi where the last two summations are over the first Em] primes.

Note: this proposition is usually used in the case that K/L is totally ramified at some prime. However it could also be used if K/L is normal with a non-abelian simple group as Galois group). Proof. Let H be the maximal unramified (at all primes) abelian extension of L. By class field theory, Gal(H/L) is isomorphic to the ideal class group of L. The assumptions on K/L imply that H n K = L. Therefore [KH: K] = [H: L]. But KH/K is unramified abelian, so is contained in the maximal unramified abelian extension of K.

Consequently, when working with eM-fields we may talk about ii, which is well-defined. Also, lal 2 = aii, if rational, is independent of the embedding. 6. For example, if c is a unit then c/s is an algebraic integer of absolute value 1, hence a root of unity. 10. Let K be a eM-field, K+ its maximal real subfield, and let hand h+ be the respective class numbers. Then h+ divides h. The quotient h- is called the relative class number (some authors call h- the first factor). Proof. We need the following result from class field theory.