Geometry of numbers

Geometry of numbers, also known as geometric number theory, is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in R n , {\displaystyle \mathbb {R} ^{n},} and the study of these lattices provides fundamental information on algebraic numbers. Hermann Minkowski (1896) initiated this line of research at the age of 26 in his work The Geometry of Numbers.

The geometry of numbers has a close relationship with other fields of mathematics, especially functional analysis and Diophantine approximation, the problem of finding rational numbers that approximate an irrational quantity.

Minkowski's results

Suppose that Γ {\displaystyle \Gamma } is a lattice in n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} and K {\displaystyle K} is a convex centrally symmetric body. Minkowski's theorem, sometimes called Minkowski's first theorem, states that if vol ⁡ ( K ) > 2 n vol ⁡ ( R n / Γ ) {\displaystyle \operatorname {vol} (K)>2^{n}\operatorname {vol} (\mathbb {R} ^{n}/\Gamma )} , then K {\displaystyle K} contains a nonzero vector in Γ {\displaystyle \Gamma } .

The successive minimum λ k {\displaystyle \lambda _{k}} is defined to be the infimum of the numbers λ {\displaystyle \lambda } such that λ K {\displaystyle \lambda K} contains k {\displaystyle k} linearly independent vectors of Γ {\displaystyle \Gamma } . Minkowski's theorem on successive minima, sometimes called Minkowski's second theorem, is a strengthening of his first theorem and states that

λ 1 λ 2 ⋯ λ n vol ⁡ ( K ) ≤ 2 n vol ⁡ ( R n / Γ ) . {\displaystyle \lambda _{1}\lambda _{2}\cdots \lambda _{n}\operatorname {vol} (K)\leq 2^{n}\operatorname {vol} (\mathbb {R} ^{n}/\Gamma ).}

Algebraic number theory Minkowski applied his results to the area of algebraic number theory, and this was one motivation for the term geometry of numbers. The ring of integers in a number field can be embedded as a lattice in a higher dimensional space. The Gaussian integers, which are all a + i b {\displaystyle a+ib} with a , b {\displaystyle a,b} integers, already is a lattice in the complex plane. Other rings of integers are not obviously lattices, like Z [ 2 ] {\displaystyle \mathbb {Z} [{\sqrt {2}}]} , which is contained in the real line, but is dense. Minkowski's basic idea was to embed numbers in a higher dimensional space, and this gives one explanation of why the general theory has been termed "the geometry of numbers". Every ring of integers can be embedded into a higher-dimensional Euclidean space in which it becomes a lattice. More generally, every fractional ideal embeds as a lattice. Estimates on the sizes of lattice vectors and volumes then lead to norm-bounds on the size of representative ideals within each ideal class. In particular, the geometry of numbers gave the first proof that the ideal class group is finite, because the number of elements in a lattice of bounded norm is finite, which was a major unsolved problem prior to Minkowski's work. Related geometric arguments supply an alternative proof of the Dirichlet unit theorem. Minkowski's construction embeds a number field K {\displaystyle K} simultaneously into all of its real and complex completions, that is, embeddings σ {\displaystyle \sigma } of K {\displaystyle K} into C {\displaystyle \mathbb {C} } . These may be real, if σ ( K ) ⊂ R {\displaystyle \sigma (K)\subset \mathbb {R} } , or complex otherwise. If K {\displaystyle K} has r 1 {\displaystyle r_{1}} real embeddings and r 2 {\displaystyle r_{2}} pairs of complex embeddings, then the Minkowski embedding realizes

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