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2018•Experimental MathematicsRequires access

On the Class Numbers in the Cyclotomic Z29- and Z31-Extensions of the Field of Rationals

Yuta Kogoshi, Takayuki Morisawa

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Abstract

Let p be a prime number. For each prime number ℓ, we consider the problem whether ℓ divides class numbers of finite subextensions in the cyclotomic Zp-extension of the field of rationals or not. Even in the case where ℓ is a primitive root modulo p2, the problem is not solved in general. In this paper, we focus on the case p = 29 and 31. And we show that ℓ does not divide class numbers of finite subextensions in the cyclotomic Z29- and Z31-extensions of the field of rationals if a prime number ℓ is a primitive root modulo p2.

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What this paper is about

Let p be a prime number. For each prime number ℓ, we consider the problem whether ℓ divides class numbers of finite subextensions in the cyclotomic Zp-extension of the field of rationals or not. Even in the case where ℓ is a primitive root modulo p2, the problem is not solved in general. In this paper, we focus on the case p = 29 and 31. And we show that ℓ does not divide class numbers of finite subextensions in the cyclotomic Z29- and Z31-extensions of the field of rationals if a prime number ℓ is a primitive root modulo p2.

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Available abstract

Let p be a prime number. For each prime number ℓ, we consider the problem whether ℓ divides class numbers of finite subextensions in the cyclotomic Zp-extension of the field of rationals or not. Even in the case where ℓ is a primitive root modulo p2, the problem is not solved in general. In this paper, we focus on the case p = 29 and 31. And we show that ℓ does not divide class numbers of finite subextensions in the cyclotomic Z29- and Z31-extensions of the field of rationals if a prime number ℓ is a primitive root modulo p2.

Key concepts: Mathematics, Rational number, Primitive root modulo n, Modulo, Prime number, Prime (order theory), Class number, Cyclotomic field

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