Number Fields Sample Clauses

Number Fields. These will be the most tangible invariants of hyperbolic manifolds. By use of PARI, we will be able to produce polynomials whose roots are eigenvalues of matrices in PSL2(C) that correspond to geodesics in arithmetic hyperbolic 3-manifolds and other polynomials that can be used to construct fields that are invariant trace fields of arithmetic hyperbolic 3-manifolds. The following material comes from Chapter 0 of [16] and [21].

Number Fields. An algebraic number field, or simply number field, is a field of finite degree over the field Q of rational numbers. In other words, a number field K is a finite extension Q(α) of Q, for some α ∈ K. Hence, K is a field that contains Q and a finite dimensional vector space over Q. The dimension of K as a vector space over Q is simply called the degree of K. A valuation | · | on K is a function from K to the non-negative real numbers such that, for all α, β ∈ K: 1. |α| = 0 if and only if α = 0, 2. |αβ| = |α||β|,