mersenneforum.org Cyclic fields with class number h
 User Name Remember Me? Password
 Register FAQ Search Today's Posts Mark Forums Read

 2018-01-30, 06:10 #1 carpetpool     "Sam" Nov 2016 5·67 Posts Cyclic fields with class number h Is it true that for any prime p = 1 (mod n), there is a polynomial P of degree n generating a cyclic field K = Q(ζ) where ζ is a root of P and K has class number p? This is necessary to study certain structures mod p. For example, p = 13, we have the vector of exponents F0=[0], F1=[1, 5, 8, 12], F2=[3, 2, 11, 10], F3=[9, 6, 7, 4]. Each of these vectors Fn represents an ideal class group. F2 and F3 are cosets of F1, and F0 is the zero group, or principal ideal class group. To find an example, one would need to find a cyclic quartic field with class number p = 13. I don't know of such an example, is someone able to give a generating polynomial for such a field?

 Similar Threads Thread Thread Starter Forum Replies Last Post carpetpool Abstract Algebra & Algebraic Number Theory 3 2018-01-13 18:13 davar55 Homework Help 67 2017-05-23 13:16 Nick Number Theory Discussion Group 0 2017-04-11 18:37 Nick Number Theory Discussion Group 0 2017-01-07 13:15 Raman Miscellaneous Math 5 2013-06-12 13:54

All times are UTC. The time now is 00:53.

Sun Nov 27 00:53:04 UTC 2022 up 100 days, 22:21, 0 users, load averages: 1.01, 1.01, 1.16

Copyright ©2000 - 2022, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.

≠ ± ∓ ÷ × · − √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °
∠ ∟ ° ≅ ~ ‖ ⟂ ⫛
≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘ ∏ ∐ ∑ ∧ ∨ ∩ ∪ ⨀ ⊕ ⊗ 𝖕 𝖖 𝖗 ⊲ ⊳
∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟
¬ ∨ ∧ ⊕ → ← ⇒ ⇐ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣ … ⋯ ⋮ ⋰ ⋱
∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ
𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎𝜍 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔