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Field theory theorem
theory, the primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element. This theorem implies
Primitive_element_theorem
Field extension generated by a one element
single element, called a primitive element. Simple extensions are well understood and can be completely classified. The primitive element theorem states
Simple_extension
Topics referred to by the same term
(free group), an element of a free generating set Primitive element (Lie algebra), a Borel-weight vector Primitive element theorem Primitive root (disambiguation)
Primitive_element
Generator of the multiplicative group of a finite field
a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF(q) is called a primitive element
Primitive element (finite field)
Primitive_element_(finite_field)
Mathematical theorem used in cryptography
with σ 1 = Id {\displaystyle \sigma _{1}={\text{Id}}} . By the primitive element theorem there exists α ∈ K {\displaystyle \alpha \in K} such that K =
Normal_basis
In algebra, a primitive element of a co-algebra C (over an element g) is an element x that satisfies μ ( x ) = x ⊗ g + g ⊗ x {\displaystyle \mu (x)=x\otimes
Primitive element (co-algebra)
Primitive_element_(co-algebra)
Theorem in algebraic geometry
simple proofs use Gauss's lemma on primitive polynomials as a main step. Primitive element theorem — another theorem asserting that certain field extensions
Lüroth's_theorem
Result due to Kummer on cyclic extensions of fields that leads to Kummer theory
'\mapsto \ell \otimes a\sigma ^{-1}(\ell ').\end{cases}}} The primitive element theorem gives L = K ( α ) {\displaystyle L=K(\alpha )} for some α {\displaystyle
Hilbert's_Theorem_90
Gives the rank of the group of units in the ring of algebraic integers of a number field
0 or r2 = 0. Other ways of determining r1 and r2 are use the primitive element theorem to write K = Q ( α ) {\displaystyle K=\mathbb {Q} (\alpha )}
Dirichlet's_unit_theorem
Algebraic structure with addition, multiplication, and division
theory from 1928 through 1942, eliminating the dependency on the primitive element theorem. A commutative ring is a set that is equipped with an addition
Field_(mathematics)
Construction of a larger algebraic field by "adding elements" to a smaller field
0, every finite extension is a simple extension. This is the primitive element theorem, which does not hold true for fields of non-zero characteristic
Field_extension
(polynomials) Polynomial remainder theorem (polynomials) Primitive element theorem (field theory) Rational root theorem (algebra, polynomials) Solutions
List_of_theorems
Theorem in mathematical logic
strengthened finite Ramsey theorem is then a computable function of n, m, k, but grows extremely fast. In particular it is not primitive recursive, but it also
Paris–Harrington_theorem
Gives information about the Galois module structure of class groups of cyclotomic fields
Stickelberger element of F {\displaystyle F} and the Stickelberger ideal of F {\displaystyle F} can be defined. By the Kronecker–Weber theorem there is an
Stickelberger's_theorem
Permutation group that preserves no non-trivial partition
\{1,\ldots ,n\}} is primitive for every n > 2. Block (permutation group theory) Jordan's theorem (symmetric group) O'Nan–Scott theorem, a classification
Primitive_permutation_group
Filtration of the Galois group of a local field extension
integers of K {\displaystyle K} . (This is stronger than the primitive element theorem.) Then, for each integer i ≥ − 1 {\displaystyle i\geq -1} , we
Ramification_group
Theorem in linear algebra
In matrix theory, the Perron–Frobenius theorem, proved in its first part by Oskar Perron (1907) and extended by Georg Frobenius (1912), asserts that a
Perron–Frobenius_theorem
Complex number that solves a monic polynomial with integer coefficients
algebraic number θ ∈ C {\displaystyle \theta \in \mathbb {C} } by the primitive element theorem. α ∈ K is an algebraic integer if there exists a monic polynomial
Algebraic_integer
Type of algebraic field extension
The equivalence of 3. and 1. is known as the primitive element theorem or Artin's theorem on primitive elements. Properties 4. and 5. are the basis of
Separable_extension
Theorem about natural numbers
In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein
Goodstein's_theorem
On prime divisors of differences two nth powers
a^{n}+b^{n}} has at least one primitive prime divisor with the exception 2 3 + 1 3 = 9 {\displaystyle 2^{3}+1^{3}=9} . Zsigmondy's theorem is often useful, especially
Zsigmondy's_theorem
Computational method
over Q {\displaystyle \mathbb {Q} } with high probability by the primitive element theorem. If this is the case, we can compute the minimal polynomial q
Factorization_of_polynomials
In mathematics, element that equals its square
mathematics, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element is idempotent under the ring's
Idempotent_(ring_theory)
Theorems that help decompose a finite group based on prime factors of its order
order of G, then there exists an element (and thus a cyclic subgroup generated by this element) of order p in G. Theorem (2)—Given a finite group G and
Sylow_theorems
On subsets of the integers in which no member of the set is a multiple of any other
\dots n\}} is called primitive if it has the property that no subset element is a multiple of any other element. Behrend's theorem states that the logarithmic
Behrend's_theorem
Limitative results in mathematical logic
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Aspect of algebraic number theory
generated over K by θ (such a θ is guaranteed to exist by the primitive element theorem), and then to examine the minimal polynomial H(X) of θ over K;
Splitting of prime ideals in Galois extensions
Splitting_of_prime_ideals_in_Galois_extensions
Describes statistically the splitting of primes in a given Galois extension of Q
mathematics, specifically in algebraic number theory, the Chebotarev density theorem, named after Nikolai Chebotarev, statistically describes the splitting
Chebotarev_density_theorem
Every set is smaller than its power set
question marks, boxes, or other symbols. In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A {\displaystyle
Cantor's_theorem
Three results in the representation theory of finite groups
Brauer's main theorems are three theorems in representation theory of finite groups linking the blocks of a finite group (in characteristic p) with those
Brauer's_three_main_theorems
Finite extension of the rationals
some element x ∈ K {\displaystyle x\in K} . By the primitive element theorem, there exists such an x {\displaystyle x} , called a primitive element. If
Algebraic_number_field
element outside this union will generate L {\displaystyle L} . This theorem was found and proven in 1910 by Ernst Steinitz. Lemma 9.19.1 (Primitive element)
Steinitz's theorem (field theory)
Steinitz's_theorem_(field_theory)
Function computable with bounded loops
primitive recursive. The Paris–Harrington theorem involves a total recursive function that is not primitive recursive. The Sudan function The Goodstein
Primitive_recursive_function
On prime divisors in Fibonacci and Lucas sequences
A246556 in the OEIS) Zsigmondy's theorem Yabuta, Minoru (2001). "A simple proof of Carmichael's theorem on primitive divisors" (PDF). Fibonacci Quarterly
Carmichael's_theorem
Mathematical theorem
Jacobson density theorem is a theorem concerning simple modules over a ring R. The theorem can be applied to show that any primitive ring can be viewed
Jacobson_density_theorem
Theorem in mathematical logic
compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important
Compactness_theorem
by a theorem of Borel–Morozov on the conjugacy of solvable subalgebras.) Given a g {\displaystyle {\mathfrak {g}}} -module V, a primitive element of V
Borel_subalgebra
Relation between sides of a right triangle
oldest extant axiomatic proof of the theorem is presented, along with Euclid's formula for generating all primitive Pythagorean triples. With contents known
Pythagorean_theorem
Fundamental combinatorial result of Ramsey theory
hypercube that is the subject of the theorem. A variable word w(x) over WH n still has length H but includes the special element x in place of at least one of
Hales–Jewett_theorem
Modular arithmetic concept
1 in the ring Z n {\displaystyle \mathbb {Z} _{n}} ), or simply a primitive element of Z n × {\displaystyle \mathbb {Z} _{n}^{\times }} . When Z n × {\displaystyle
Primitive_root_modulo_n
Equivalence of notions of density for sets of multiples of integers
Khachatrian, Levon H. (1997), "Classical results on primitive and recent results on cross-primitive sequences: Theorem 1.11", The Mathematics of Paul Erdős, I, Algorithms
Davenport–Erdős_theorem
Theorem in set theory
In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there
Schröder–Bernstein_theorem
Standard system of axiomatic set theory
Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in
Zermelo–Fraenkel_set_theory
Theorem for proving more complex theorems
also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however
Lemma_(mathematics)
Existence and cardinality of models of logical theories
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf
Löwenheim–Skolem_theorem
Function in mathematical number theory
abelian group, there must exist an element whose order equals the exponent, λ(n). Such an element is called a primitive λ-root modulo n. The Carmichael function
Carmichael_function
Undecidability of equality of real numbers
by other primitives than in Richardson's theorem, there exist algorithms that can determine whether an expression is zero. Richardson's theorem can be stated
Richardson's_theorem
Field theory is the branch of algebra that studies fields
single element, called a primitive element, or generating element. The primitive element theorem classifies such extensions. Normal extension An extension
Glossary_of_field_theory
Algebraic structure
Fermat's little theorem. If a {\displaystyle a} is a primitive element in G F ( q ) {\displaystyle \mathrm {GF} (q)} , then for any non-zero element x {\displaystyle
Finite_field
polynomial, and lifting the result to a factorization of the primitive part. Rational root theorem B. Hartley; T.O. Hawkes (1970). Rings, modules and linear
Primitive_part_and_content
Symbol representing a property or relation in logic
{\displaystyle a} and b {\displaystyle b} . Predicates are considered a primitive notion of first-order and higher-order logic, and are therefore not defined
Predicate_(logic)
Axiom of set theory
{\displaystyle {\mathcal {A}}} has at least one element maximal with respect to inclusion. König's theorem: Informally, the sum of a sequence of cardinals
Axiom_of_choice
In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations (specifically in the representation
Weyl's theorem on complete reducibility
Weyl's_theorem_on_complete_reducibility
Subfield of automated reasoning and mathematical logic
proof for a theorem is certified valid. For this, it is generally required that each individual proof step can be verified by a primitive recursive function
Automated_theorem_proving
Number with an integer power equal to 1
{(z+1)^{n}-1}{(z+1)-1}},} and expanding via the binomial theorem. Every nth root of unity is a primitive dth root of unity for exactly one positive divisor
Root_of_unity
Mathematical proposition equivalent to the axiom of choice
the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space
Zorn's_lemma
Group whose operation is composition of permutations
Each element of G can be thought of as a permutation in this way and so G is isomorphic to a permutation group; this is the content of Cayley's theorem. For
Permutation_group
Concept in computability theory
Kleene's normal form theorem for computable functions (Soare 1987, p. 15; Kleene 1943, p. 52—53). This states there exists a fixed primitive recursive function
Kleene's_T_predicate
Function that preserves distinctness
monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see Homomorphism § Monomorphism
Injective_function
in the theorem is induced from an irreducible character of the inertial subgroup IG(μ). If, for example, the irreducible character χ is primitive (that
Clifford_theory
About products of primitive polynomials
irreducible in Q[X] and primitive in Z[X]. The proof is given below for the more general case. Note that an irreducible element of Z (a prime number) is
Gauss's_lemma_(polynomials)
Theorem that arithmetical truth cannot be defined in arithmetic
metalanguage includes primitive notions, axioms, and rules of inference absent from the object language, so that there are theorems provable in the metalanguage
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Set of sentences in a formal language
deduction rules. An element ϕ ∈ T {\displaystyle \phi \in T} of a deductively closed theory T {\displaystyle T} is then called a theorem of the theory. In
Theory_(mathematical_logic)
Mathematical construction
include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization
Ultraproduct
branch of abstract algebra called ring theory, the double centralizer theorem can refer to any one of several similar results. These results concern
Double_centralizer_theorem
Study of computable functions and Turing degrees
by Post's theorem. A weaker relationship was demonstrated by Kurt Gödel in the proofs of his completeness theorem and incompleteness theorems. Gödel's
Computability_theory
Numerical method for solving physical or engineering problems
Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical
Finite_element_method
Algebraic structure
to divisibility: any element of a PID has a unique factorization into prime elements (so an analogue of the fundamental theorem of arithmetic holds);
Principal_ideal_domain
Algebraic structure
Jacobson density theorem is a theorem concerning simple modules over a ring R. The theorem can be applied to show that any primitive ring can be viewed
Noncommutative_ring
Periodic set of points
} A primitive element of a lattice is an element v ∈ Λ {\displaystyle v\in \Lambda } that is not a positive integer multiple of another element in the
Lattice_(group)
Axioms for the natural numbers
Foundations of mathematics Frege's theorem Goodstein's theorem Neo-logicism Non-standard model of arithmetic Paris–Harrington theorem Presburger arithmetic Skolem
Peano_axioms
Theorem in set theory
In set theory, Kőnig's theorem states that if the axiom of choice holds, I is a set, κ i {\displaystyle \kappa _{i}} and λ i {\displaystyle \lambda _{i}}
Kőnig's_theorem_(set_theory)
theorem Goodstein's theorem Green's theorem (to do) Green's theorem when D is a simple region Heine–Borel theorem Intermediate value theorem Itô's lemma Kőnig's
List_of_mathematical_proofs
Area of mathematical logic
It's a consequence of Gödel's completeness theorem (not to be confused with his incompleteness theorems) that a theory has a model if and only if it
Model_theory
3-volume treatise on mathematics, 1910–1913
ideas and methods of mathematical logic and to minimise the number of primitive notions, axioms, and inference rules; to precisely express mathematical
Principia_Mathematica
Basic framework of mathematics
organizing a field of knowledge by means of primitive concepts, axioms, postulates, definitions, and theorems. Aristotle took a majority of his examples
Foundations_of_mathematics
Mathematical logic concept
arithmetic and that its consistency is therefore less controversial. Gentzen's theorem is concerned with first-order arithmetic: the theory of the natural numbers
Gentzen's_consistency_proof
Indefinite integral
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable
Antiderivative
Fundamental theorem in mathematical logic
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability
Gödel's_completeness_theorem
Mathematical set of all subsets of a set
power set must be larger than the original set). In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite
Power_set
Sufficient condition for a separable extension of a Hilbertian field to be Hilbertian
using the diamond theorem If L is finite over K, it is Hilbertian; hence we assume that L/K is infinite. Let x be a primitive element for L/N, i.e., L
Haran's_diamond_theorem
Equations of degree 5 or higher cannot be solved by radicals
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial
Abel–Ruffini_theorem
As proven by Tarski, this theory is decidable; see Tarski–Seidenberg theorem and Quantifier elimination. Current implementations of decision procedures
Decidability of first-order theories of the real numbers
Decidability_of_first-order_theories_of_the_real_numbers
Any one of the distinct objects that make up a set in set theory
In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called A containing
Element_of_a_set
Axiom set used in first-order logic
require an underlying set theory. The only primitive objects of the system are "points" and the only primitive predicates are "betweenness" (expressing
Tarski's_axioms
\omega } has a primitive defined in a neighborhood of every point of M {\displaystyle M} . For a global generalization of the fundamental theorem, one needs
Darboux_derivative
One-to-one correspondence
function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Given
Bijection
Symbol representing a mathematical concept
be defined over the whole domain of discourse. Function symbols are a primitive notion, and are therefore not defined in terms of other, more basic concepts
Function_symbol
Statement that is taken to be true
assertions (axioms, postulates, propositions, theorems) and definitions. One must concede the need for primitive notions, or undefined terms or concepts, in
Axiom
Branch of mathematics that studies algebraic structures
density theorem Wedderburn's little theorem Lasker–Noether theorem Field (mathematics) Subfield (mathematics) Multiplicative group Primitive element (field
List of abstract algebra topics
List_of_abstract_algebra_topics
Type of module over a ring
subset of U. Then there exists an element r of R such that x⋅A = x⋅r for all x in X. In particular, any primitive ring may be viewed as (that is, isomorphic
Simple_module
Mathematical connection between field theory and group theory
between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group
Galois_theory
Axiomatic logical system
induction present in arithmetics stronger than Q turns this axiom into a theorem. x + 0 = x x + Sy = S(x + y) (4) and (5) are the recursive definition of
Robinson_arithmetic
System of mathematical set theory
There is a single primitive binary relation, set membership; that set a is a member of set b is written a ∈ b (usually read "a is an element of b"). The symbolic
General_set_theory
Polynomial without nontrivial factorization
factorization domain the same theorem is true, but is more accurately formulated by using the notion of primitive polynomial. A primitive polynomial is a polynomial
Irreducible_polynomial
approach called domain theory, where they are considered as a kind of primitive element: the information represented by compact elements cannot be obtained
Compact_element
Mathematical set that can be enumerated
injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements
Countable_set
Theory of truth in the philosophy of language
notably Tarski's undefinability theorem using the same formal technique Kurt Gödel used in his incompleteness theorems. Roughly, this states that a truth-predicate
Semantic_theory_of_truth
Type of logical system
to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem. First-order logic is the standard for the formalization
First-order_logic
Arithmetic in a field with a finite number of elements
that x is a primitive element. There is at least one irreducible polynomial for which x is a primitive element. In other words, for a primitive polynomial
Finite_field_arithmetic
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