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Power integral bases in prime-power cyclotomic fields

✍ Scribed by István Gaál; Leanne Robertson

Publisher
Elsevier Science
Year
2006
Tongue
English
Weight
152 KB
Volume
120
Category
Article
ISSN
0022-314X

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✦ Synopsis

Let p be an odd prime and q = p m , where m is a positive integer. Let ζ be a primitive qth root of unity, and O q be the ring of integers in the cyclotomic field Q(ζ ). We prove that if O q = Z[α] and gcd(h + q , p(p -1)/2) = 1, where h + q is the class number of Q(ζ + ζ -1 ), then an integer translate of α lies on the unit circle or the line Re(z) = 1/2 in the complex plane. Both are possible since

We conjecture that, up to integer translation, these two elements and their Galois conjugates are the only generators for O q , and prove that this is indeed the case when q = 25.

📜 SIMILAR VOLUMES

Power Bases for 2-Power Cyclotomic Field
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Let `be a primitive 2 m th root of unity. We prove that Z[:]=Z [`] if and only if :=n\`i for some n, i # Z, i odd. This is the first example of number fields of arbitrarily large degree for which all power bases for the ring of integers are known. 2001 Academic Press ## 1. Introduction A number f

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Let p be an odd prime and O p be the ring of integers in the cyclotomic field Q(`), where `is a primitive p th root of unity. Then O p =Z[:] if :=`, 1Â(1+`), or one of the conjugates of these two elements. In 1988, Bremner [3] conjectured that up to integer translation there are no further generator