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- Abelian variety : A complete non-singular projective variety which is also an algebraic group, necessarily abelian; a complex torus.
- Algebra over a field : A ring containing an isomorphic copy of a given field in its centre.
- Algebraic independence : The property of elements of an extension field which satisfy only the trivial polynomial relation.
- Algebraic number field : A field extension of the rational numbers of finite degree; a principal object of study in algebraic number theory.
- Algebraic number : A complex number that is a root of a polynomial with rational coefficients.
- Centraliser : The set of all group elements which commute with every element of a given subset.
- Centre of a ring : The subring of a ring consisting of all elements which commute with every element of the ring.
- Complement (linear algebra) : A pair of subspaces which form an (internal) direct sum.
- Conductor of an abelian variety : A measure of the nature of the bad reduction at some prime.
- Conjugation (group theory) : The elements of any group that may be partitioned into conjugacy classes.
- Content (algebra) : The highest common factor of the coefficients of a polynomial.
- Cyclotomic field : An algebraic number field generated over the rational numbers by roots of unity.
- Derivation (mathematics) : A map defined on a ring which behaves formally like differentiation: D(x.y)=D(x).y+x.D(y).
- Discriminant of a polynomial : An invariant of a polynomial which vanishes if it has a repeated root: the product of the differences between the roots.
- Division ring : (or skew field), In algebra it is a ring in which every non-zero element is invertible.
- Exact sequence : A sequence of algebraic objects and morphisms which is used to describe or analyse algebraic structure.
- 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.
- Group (mathematics) : Set with a binary associative operation such that the operation admits an identity element and each element of the set has an inverse element for the operation.
- Group theory : Branch of mathematics concerned with groups and the description of their properties.
- Idempotence : The property of an operation that repeated application has no effect.
- Inner product : A bilinear or sesquilinear form on a vector space generalising the dot product in Euclidean spaces.
- Integral domain : A commutative ring in which the product of two non-zero elements is again non-zero.
- Linear independence : The property of a system of elements of a module or vector space, that no non-trivial linear combination is zero.
- Local ring : A ring with a unique maximal ideal.
- Localisation (ring theory) : An extension ring in which elements of the base ring become invertible.
- Manin obstruction : A measure of the failure of the Hasse principle for geometric objects.
- Monoid : An algebraic structure with an associative binary operation and an identity element.
- Noetherian module : Module in which every ascending sequence of submodules has only a finite number of distinct members.
- Noetherian ring : A ring satisfying the ascending chain condition on ideals; equivalently a ring in which every ideal is finitely generated.
- Normaliser : The elements of a group which map a given subgroup to itself by conjugation.
- Polynomial ring : Ring formed from the set of polynomials in one or more variables with coefficients in another ring.
- Resolution (algebra) : An exact sequence which is used to describe the structure of a module.
- Resultant (algebra) : An invariant which determines whether or not two polynomials have a factor in common.
- S-unit : An element of an algebraic number field which has a denominator confined to primes in some fixed set.
- Splitting field : A field extension generated by the roots of a polynomial.
- Stably free module : A module which is close to being free: the direct sum with some free module is free.
- Symmetric group : The group of all permutations of a set, that is, of all invertible maps from a set to itself.
- Szpiro's conjecture : A relationship between the conductor and the discriminant of an elliptic curve.
- Weierstrass preparation theorem : A description of a canonical form for formal power series over a complete local ring.