page_title: Dirichlet's unit theorem
section_title: Examples
hierarchical_section_title: Dirichlet's unit theorem / The regulator / Examples
caption_attribution_description: English: A fundamental domain in logarithmic space of the group of units of the number field K obtained by adjoining to Q a root of f(x)&#160;=&#160;x3&#160;+&#160;x2&#160;&#8722;&#160;2x&#160;&#8722;&#160;1. This field is a Galois extension of degree 3. Its discriminant is 49&#160;=&#160;72. If &#945; denotes the root of f(x) which is approximately 1.24698, then a set of fundamental units is {ε1,&#160;ε2} where ε1&#160;=&#160;α2&#160;+&#160;α&#160;&#8722;&#160;1 and ε2&#160;=&#160;2&#160;&#8722;&#160;α2. The fundamental domain is spanned by v1 and v2 with the ith component of vj equal to log&#x7c;&#963;i(εj)&#x7c;. So, approximately, v1&#160;=&#160;(0.588863,&#160;&#8722;0.809587,&#160;0.220724) and v1&#160;=&#160;(&#8722;0.809587,&#160;0.220724,&#160;0.588863). The area of the fundamental domain is approximately 0.910114, so the regulator of K is approximately 0.525455. See Table B.4 of Cohen's A course in computational algebraic number theory (1993). Computed using PARI/GP. Plotted using Mathematica.
context_page_description: In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring OK of algebraic integers of a number field K. The regulator is a positive real number that determines how "dense" the units are.
  The statement is that the group of units is finitely generated and has rank equal to
  r = r₁ + r₂ − 1
  where r₁ is the number of real embeddings and r₂ the number of conjugate pairs of complex embeddings of K. This characterisation of r₁ and r₂ is based on the idea that there will be as many ways to embed K in the complex number field as the degree n = [K: ℚ]; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that
  n = r₁ + 2r₂.
  Note that if K is Galois over ℚ then either r₁=0 or r₂=0.
  Other ways of determining r₁ and r₂ are
  use the primitive element theorem to write K = ℚ, and then r₁ is the number of conjugates of α that are real, 2r₂ the number that are complex; in other words, if f is the minimal polynomial of α over ℚ, then r₁ is the number of real roots and 2r₂ is the number of non-real complex roots of f;
context_section_description: The regulator of an imaginary quadratic field, or of the rational integers, is 1 (as the determinant of a 0 × 0 matrix is 1).
  The regulator of a real quadratic field is the logarithm of its fundamental unit: for example, that of ℚ(√5) is log . This can be seen as follows. A fundamental unit is , and its images under the two embeddings into ℝ are  and . So the r × (r + 1) matrix is
  The regulator of the cyclic cubic field ℚ(α), where α is a root of x³ + x² − 2x − 1, is approximately 0.5255. A basis of the group of units modulo roots of unity is {ε₁, ε₂} where ε₁ = α² + α − 1 and ε₂ = 2 − α².
caption: A fundamental domain in logarithmic space of the group of units of the cyclic cubic field K obtained by adjoining to ℚ a root of f(x) = x3 + x2 − 2x − 1. If α denotes a root of f(x), then a set of fundamental units is {ε1, ε2}, where ε1 = α2 + α − 1 and ε2 = 2 − α2. The area of the fundamental domain is approximately 0.910114, so the regulator of K is approximately 0.525455.
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