THE CONJUGATE DIMENSION OF ALGEBRAIC NUMBERS

We find sharp upper and lower bounds for the degree of an algebraic number in terms of the Q-dimension of the space spanned by its conjugates. For all but seven non-negative integers n the largest degree of an algebraic number whose conjugates span a vector space of dimension n is equal to 2 n n!. The proof, which covers also the seven exceptional cases, uses a result of Feit on the maximal order of finite subgroups of GL n(Q); this result depends on the classification of finite simple groups. In particular, we construct an algebraic number of degree 1152 whose conjugates span a vector space of dimension only 4. We extend our results in two directions. We consider the problem when Q is replaced by an arbitrary field, and prove some general results. In particular, we again obtain sharp bounds when the ground field is a finite field, or a cyclotomic extension Q(ω� ) of Q. Also, we look at a multiplicative version of the problem by considering the analogous rank problem for the multiplicative group generated by the conjugates of an algebraic number.

[1]  C. Smyth,et al.  Additive and multiplicative relations connecting conjugate algebraic numbers , 1986 .

[2]  Jean-Pierre Serre,et al.  Topics in Galois Theory , 1992 .

[3]  James Kuzmanovich,et al.  Finite Groups of Matrices Whose Entries Are Integers , 2002, Am. Math. Mon..

[4]  H. S. M. Coxeter,et al.  The product of the generators of a finite group generated by reflections , 1951 .

[5]  Andrzej Schinzel,et al.  Polynomials with Special Regard to Reducibility , 2000 .

[6]  C. Chevalley.  Invariants of Finite Groups Generated by Reflections , 1955 .

[7]  S. D. Chatterji.  Proceedings of the International Congress of Mathematicians , 1995 .

[8]  Nicolas Bourbaki,et al.  Groupes et algèbres de Lie , 1971 .

[9]  Wilhelm Ljunggren,et al.  On the Irreducibility of Certain Trinomials and Quadrinomials. , 1960 .

[10]  Larry Smith,et al.  Polynomial Invariants of Finite Groups , 1995 .

[11]  K. Brown.  POLYNOMIAL INVARIANTS OF FINITE GROUPS (Research Notes in Mathematics 6) By Larry Smith: 360 pp., US$59.95, ISBN 1 56881 053 9 (A. K. Peters, 1995) , 1997 .

[12]  B. Poonen.  Points Having the Same Residue Field as Their Image under a Morphism , 2001 .

[13]  David Goss,et al.  Basic Structures of Function Field Arithmetic , 1997 .

[14]  R. Carter.  REFLECTION GROUPS AND COXETER GROUPS (Cambridge Studies in Advanced Mathematics 29) , 1991 .

[15]  P. Gallagher.  Invariants for finite groups , 1979 .

[16]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[17]  D. J. H. Garling,et al.  A Course in Galois Theory , 1987 .

[18]  M. Mehta.  Basic sets of invariant polynomials for finite reflection groups , 1988 .

[19]  Michael Francis Atiyah,et al.  Introduction to commutative algebra , 1969 .

[20]  J. Humphreys.  Reflection groups and coxeter groups , 1990 .

[21]  G. C. Shephard,et al.  Finite Unitary Reflection Groups , 1954, Canadian Journal of Mathematics.

[22]  J. S. Frame.  The classes and representations of the groups of 27 lines and 28 bitangents , 1951 .