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A finite degree extension of the field of rational numbers Q. Such a field can always be written as Q(a), that is, has a basis as a vector space over Q of the form 1,a,a2,...,ad-1 where d is the degree.
A quadratic field is of degree 2, a cubic field is of degree 3, quartic of degree 4.
The term biquadratic field used to be an alternative for quartic, but more recently has the restricted meaning of a quartic field that is composed of three quadratic subfields
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An element of an algebraic number field: an element, a, of a field containing Q which satisfies a polynomial equation with rational coefficients. The minimal equation is the polynomial of smallest degree which a satisfies, taken to be monic, that is, to have leading coefficient 1.
The field Q(a) is then an algebraic number field.
The set of all complex algebraic numbers forms a subfield A of C.
Complex numbers which are not algebraic, such as e and pi, are transcendental
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An element of a ring containing Z which satisfies a monic polynomial with integer coefficients.
Examples: (1 + sqrt(5))/2 is a root of X2 - X - 1.
The algebraic integers in an algebraic number field are closed under addition and multiplication, so form a ring, the maximal order.
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The ring of all algebraic integers in an algebraic number field.
Every quadratic field can be written as Q(sqrt(d)) with d a square-free integer. The ring of integers is then Z[a] where a = (1+sqrt(d))/2 if d is congruent to 1 modulo 4 and a = sqrt(d) otherwise.
A field is monogenic if the ring of integers is of the form Z[a] for some element a.
All quadratic fields are monogenic, but not all fields of higher degree.
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The units of the ring of integers of a number field of degree n and signature n=r+2s form a finitely generated abelian group with a torsion-free component of rank r+s-1. The torsion subgroup is just the subgroup of roots of unity.
The fields with unit rank zero are the rationals and imaginary quadratic fields.
The fields with unit rank one are the real quadratic fields and cubic fields with a single real embedding. A generator of the torsion-free part of the unit group is a fundamental unit.
The fundamental unit of a real quadratic field of discriminant d determines the fundamental solution to the corresponding Pell equation
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A number field K of degree n over Q will have r embeddings into the real numbers and s pairs of embeddings into the complex numbers (which are not into the reals). If K is generated by a single element x, these correspond to the real and complex conjugate pairs of roots of the minimal equation for x.
The signature is the pair (r,s).
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