## On the minimum of the unit lattice.(English)Zbl 0755.11034

Let $$K$$ be a finite extension of $$\mathbb{Q}$$ of degree $$n$$ with maximal order $$R$$. As usual, let $$r_ 1$$ and $$2r_ 2$$ be the number of real and non- real embeddings of $$K$$ into $$\mathbb{C}$$, and let $$r=r_ 1+r_ 2-1$$ be the rank of the unit group $$R^*$$. Let Log denote the standard mapping of $$K^*$$ into $$\mathbb{R}^ r$$ so that $$L:=\text{Log}(R^*)$$ is a lattice of rank $$r$$. Let $$\lambda(L)$$ denote the minimum of the Euclidean norms of the non-zero vectors in $$L$$. Using results of C. J. Smyth [Bull. Lond. Math. Soc. 3, 169-175 (1971; Zbl 0235.12003)], and E. Dobrowolski [Acta Arith. 34, 391-401 (1979; Zbl 0416.12001)], the author proves the following lower bound: $\lambda(L)>{1\over\sqrt{r+1}}{1\over1000}\left({\log\log n\over\log n}\right)^ 3.$

### MSC:

 11R27 Units and factorization 11H06 Lattices and convex bodies (number-theoretic aspects)

### Citations:

Zbl 0235.12003; Zbl 0416.12001
Full Text:

### References:

 [1] Buchmann, ZurKomplexität der Berechnung von Einheiten und Klassenzahlen algebraicher Zahlkörper, HabilitationsschriftDüsseldorf (1987). [2] Buchmann, Kessler, Computing a reduced lattice basis from a generating system, to appear. [3] Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta arithmetica34 (1979), 391-401. · Zbl 0416.12001 [4] SchinzelZassenhaus, A refinement of two theorems of Kronecker, Mich. Math. J.12 (1965), 81-84. · Zbl 0128.03402 [5] SmythOn the product of the conjugates outside the unit circle of an algebraic integer, Bull. London Math. Soc.3 (1971), 169-175. · Zbl 0235.12003
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