Math 254a: Interlude on Kummer Theory
نویسنده
چکیده
h = h− · h where h is the class number of Q(ζp), h + is the class number of Q(ζp + ζ̄p), and h ∈ N. We now consider the problem of showing that if p divides h, it must divide h. We remark that in fact it is a conjecture of Vandiver that p never divides the factor h, and this has been checked for primes into the millions. Washington [2, pp. 158-129] produces heuristics however showing that without any assumptions on p tending not to divide h, one would only expect the number of p less than 4 million such that p divides h to be roughly 1.36. Thus, he views the (superficially very substantial) numerical evidence for the conjecture to be unconvincing. We will prove:
منابع مشابه
Math 254a: First Lecture
In this course, we will cover the basics of what is called algebraic number theory. Just as number theory is often described as the study of the integers, algebraic number theory may be loosely described as the study of certain subrings of fields K with [K : Q] < ∞; these rings tend to act as natural generalizations of the integers. However, although algebraic number theory has evolved into a s...
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