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- Algebraic number field : A field extension of the rational numbers of finite degree; a principal object of study in algebraic number theory.
- Artin-Schreier polynomial : A type of polynomial whose roots generate extensions of degree p in characteristic p.
- Complex number : Numbers of the form a+bi, where a and b are real numbers and i denotes a number satisfying .
- Conductor of a number field : Used in algebraic number theory; a modulus which determines the splitting of prime ideals.
- Cyclotomic field : An algebraic number field generated over the rational numbers by roots of unity.
- Discriminant of an algebraic number field : An invariant attached to an extension of algebraic number fields which describes the geometric structure of the ring of integers and encodes ramification data.
- Elliptic curve : An algebraic curve of genus one with a group structure; a one-dimensional abelian variety.
- Field (mathematics) : An algebraic structure with operations generalising the familiar concepts of real number arithmetic.
- Field automorphism : An invertible function from a field onto itself which respects the field operations of addition and multiplication.
- Field theory (mathematics) : A subdiscipline of abstract algebra that studies fields, which are mathematical constructs that generalize on the familiar concepts of real number arithmetic.
- Galois theory : Algebra concerned with the relation between solutions of a polynomial equation and the fields containing those solutions.
- Matroid : Structure that captures the essence of a notion of 'independence' that generalizes linear independence in vector spaces.
- Minimal polynomial : The monic polynomial of least degree which a square matrix or endomorphism satisfies.
- Normal extension : A field extension which contains all the roots of an irreducible polynomial if it contains one such root.
- Quadratic equation : An equation of the form ax2 + bx + c = 0 where a, b and c are constants.
- Quadratic field : A field which is an extension of its prime field of degree two.
- Splitting field : A field extension generated by the roots of a polynomial.