Relative Pólya group and Pólya dihedral extensions of Q

Abstract

A number field with trivial Pólya group [2] is called a Pólya field. We define "relative Pólya group Po(L/K)" for L/K a finite extension of number fields, generalizing the Pólya group. Using cohomological tools in [1], we compute some relative Pólya groups. As a consequence, we generalize Leriche's results in [17] and prove the triviality of relative Pólya group for the Hilbert class field of K. Then we generalize our previous results [19] on Pólya S 3extensions of Q to dihedral extensions of Q of order 2l, for l an odd prime. We also improve Leriche's upper bound in [16] on the number of ramified primes in Pólya D l -extensions of Q and prove that for a real (resp. imaginary) Pólya D l -extension of Q at most 4 (resp. 2) primes ramify.

Ostrowski quotients for finite extensions of number fields

“…On the other hand Leriche proved that the Hilbert class field H(K) of K is a Pólya field [12, Corollary 3.2], which recently has been generalized to the triviality of Po(H(K)/K), see [15,Corollary 2.9]. Using the same method as in [15, §2] we have: Theorem 3.8.…”

“…Remark 1.4. In [15] we called (1.2) "Zantema's exact sequence", however a version of this sequence can be seen in [1] and therefore we call a genarlization of this exact sequence the "Brumer-Rosen-Zantema" exact sequence or shortly "BRZ", see Theorem (2.2) below.…”

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For a number field L, the Pólya group of L is the subgroup Po(L) of the ideal class group of L generated by all classes of Ostrowski ideals of L [17]. Relativizing this notion, the relative Pólya group Po(L/K), for L/K a finite extension of number fields, has been recently introduced independently in [3,15]. Assuming L/K is Galois, we find some applications of an exact sequence in [15] for the capitulation problem, including a new proof of Hilbert's theorem 94.

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“…Relativizing this notion, the relative Pólya group Po(L/K), for L/K a finite extension of number fields, has been recently introduced independently in [3,15]. Assuming L/K is Galois, we find some applications of an exact sequence in [15] for the capitulation problem, including a new proof of Hilbert's theorem 94. Then we define the Ostrowski quotient Ost(L/K) and generalize some known results for P o(L) to Ost(L/K).…”

Totally real bi-quadratic fields with large Pólya groups

“…From Proposition 1, it is clear that the order of the Pólya group for a quadratic field is necessarily 1 or a power of 2. In [16], Zantema generalized this to finite Galois extensions K/Q and recently, Maarefparvar and Rajaei extended the result for relative extensions of number fields in [11]. Leriche addressed the cubic field case in [8].…”