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The totally real \(A_ 6\) extension of degree 6 with minimum discriminant. (English) Zbl 0799.11052

Summary: The totally real algebraic number field \(F\) of degree 6 with Galois group \(A_ 6\) and minimum discriminant is determined. It is unique up to isomorphy and is generated by a root of the polynomial \(t^ 6- 24 t^ 4+ 21t^ 2+ 9t +1\) over the rationals. The authors also give an integral basis and list the fundamental units and class number of \(F\).
In [ibid. 1, 231-235 (1992; Zbl 0773.11067)] the authors gave details of a computation to determine the (unique) totally real algebraic number field of degree 6 having Galois group \(A_ 5\) and minimum discriminant. There they indicated how the same computation could be extended to give the totally real sextic field with Galois group \(A_ 6\) of minimum discriminant. This computation has now been completed.

MSC:

11R29 Class numbers, class groups, discriminants
11R21 Other number fields
11R80 Totally real fields
11Y40 Algebraic number theory computations

Citations:

Zbl 0773.11067
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References:

[1] Ford D., Experimental Math. 1 pp 231– (1992) · Zbl 0773.11067
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