The Minimum Discriminant of Totally Real Algebraic Number Fields of Degree 9 with Cubic Subfields

1993 ◽  
Vol 60 (202) ◽  
pp. 801
Author(s):  
Hiroyuki Fujita
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Totally Real

Finding fundamental units in algebraic number fields

1985 ◽  
Vol 98 (3) ◽  
pp. 383-388
Author(s):  
Günter Lettl
Keyword(s):  
Algebraic Number ◽  
Short Proof ◽  
General Theorem ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Successive Minima ◽  
Group Of Units ◽  
Rank 2 ◽  
Totally Real ◽  
Fundamental Units

Recently Cusick [4] presented a very elegant and short proof of the fact that a pair of fundamental units of a totally real cubic or quartic number field can be found by taking ‘successive minima’ of the function tr(ε2), where ε runs through the group of units and tr denotes the absolute trace. Hidden in Cusick's proof there is a general theorem, which shows how strictly convex functions can be used to find lattice-vectors extensible to a basis of a given geometrical lattice, and which we state and prove in § 3. A result analogous to Cusick's for some families of functions related to tr (ε2) is given in Theorem 1, thereby improving results of Brunotte and Halter-Koch [2], [5]. For a survey of unit groups of rank 2 and more literature the reader is also referred to [2].

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