Biology

A field extension that is both normal and separable.

A branch of mathematics focused on abstract structures and relationships, rather than numerical values or quantities.

A set with two binary operations (addition and multiplication) that satisfy certain axioms, including distributivity of multiplication over addition.

The number of elements in a group (if finite).

The set of all eigenvalues of a matrix.

The determinant of a square matrix is a scalar value that can be computed from the elements of the matrix.

The characteristic polynomial of a matrix A is det(A - λI), where I is the identity matrix.

A mapping between two groups that preserves the group operation.

A subspace of a vector space V is a subset of V that is itself a vector space under the same operations as V.

A polynomial that has distinct roots.

The eigenvalue associated with an eigenvector v such that Av = λv.

The minimal polynomial of a matrix A is the monic polynomial of least degree such that m(A) = 0.

A field extension E/F such that every element of E is a root of a separable polynomial in F[x].

A vector space is a set with operations of addition and scalar multiplication that satisfy certain axioms.

The smallest field extension of F that contains all the roots of a given polynomial in F[x].

A subgroup N of a group G is normal if gN = Ng for all g in G.

A commutative ring with a multiplicative identity (1) and in which every non-zero element has a multiplicative inverse.

The set of all vectors that are the image of some vector under a linear transformation.

A basis of a vector space V is a set of linearly independent vectors that spans V.

The set of elements in a group that map to the identity element in another group under a homomorphism.

An extension field E of F is an algebraic extension if every element of E is algebraic over F.

The set of all vectors that are mapped to the zero vector by a linear transformation.

A matrix A is diagonalizable if there exists an invertible matrix P such that P^{-1}AP is a diagonal matrix.

A homomorphism that is both injective (one-to-one) and surjective (onto).

An element α in an extension field E of a field F is algebraic over F if α is a root of some non-zero polynomial in F[x].

A linear transformation is a mapping between two vector spaces that preserves vector addition and scalar multiplication.

A group in which the operation is commutative (i.e., a * b = b * a for all elements a and b).

A set of vectors that spans a vector space V if every vector in V can be written as a linear combination of the vectors in the set.

The matrix representation of a linear transformation depends on the choice of bases for the vector spaces.

The group formed by the set of cosets of a normal subgroup N in G, with the operation (aN)(bN) = (ab)N.

Vectors v1, v2, ..., vn are linearly independent if no non-trivial linear combination of them equals the zero vector.

The number of vectors in a basis of a vector space.

An eigenvector of a matrix A is a non-zero vector v such that Av = λv for some scalar λ.

A polynomial with coefficients in a field.

The group of automorphisms of a Galois extension E/F that fix F.

A field that has no non-trivial algebraic extensions.

An automorphism of a field E is an isomorphism from E to itself.

A ring in which multiplication is commutative.

A collection of elements with a defined operation that satisfies certain axioms (closure, associativity, identity element, and inverse element).

A subgroup of a group G is a subset of G that is itself a group under the same operation as G.