hcju/proofwikips huggingface dataset:
Definition:Quaternion
A '''quaternion''' is a number in the form: : $a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ where: : $a, b, c, d$ are real numbers : $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ are entities related to each other in the following way: {{begin-eqn}}      {{eqn | l = \mathbf i \mathbf j = -\mathbf j \mathbf i       | r = \mathbf k       | c =   }} {{eqn | l = \mathbf j \mathbf k = -\mathbf k \mathbf j       | r = \mathbf i       | c =  }} {{eqn | l = \mathbf k \mathbf i = -\mathbf i \mathbf k       | r = \mathbf j       | c =  }} {{eqn | l = \mathbf i^2 = \mathbf j^2 = \mathbf k^2 = \mathbf i \mathbf j \mathbf k       | r = -\mathbf 1       | c =  }} {{end-eqn}}
Definition:Conjugate Quaternion
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion. The conjugate quaternion of $\mathbf x$ is defined as: :$\overline {\mathbf x} = a \mathbf 1 - b \mathbf i - c \mathbf j - d \mathbf k$. === Matrix Form === {{:Definition:Conjugate Quaternion/Matrix Form}} === Ordered Pair of Complex Numbers === {{:Definition:Conjugate Quaternion/Ordered Pair}}
Definition:Subsemiring
Let $\left({S, *, \circ}\right)$ be an algebraic structure with two operations. A '''subsemiring of $\left({S, *, \circ}\right)$''' is a subset $T$ of $S$ such that $\left({T, *_T, \circ_T}\right)$ is a semiring.  6corsnq5jtxaxkicgktu8idhrahdz68
Definition:Additive Group of Ring
The group $\struct {R, +}$ is known as the '''additive group of $R$'''.
Definition:Additive Subgroup
Let $\struct {R, +, \times}$ be a ring. Let $\struct {R, +}$ be the additive group of $\struct {R, +, \times}$. Let $\struct {S, +}$ be a subgroup of $\struct {R, +}$ Then $\struct {S, +}$ is an '''additive subgroup of ''' $\struct {R, +, \times}$.
Definition:Generated Subring
Let $\struct {R, +, \circ}$ be a ring. Let $S \subseteq R$ be a subset. The '''subring generated by $S$''' is the smallest subring of $R$ containing $S$; that is, it is the intersection of all subrings of $R$ containing $S$.
Definition:Proper Zero Divisor
Let $\struct {R, +, \circ}$ be a ring. A proper zero divisor of $R$ is an element $x \in R^*$ such that: :$\exists y \in R^*: x \circ y = 0_R$ where $R^*$ is defined as $R \setminus \set {0_R}$. That is, it is a zero divisor of $R$ which is '''specifically not $0_R$'''.
Definition:Perfect Graph
A graph is '''perfect''' if no two vertices have the same degree.
Definition:Term of Sequence
The elements of a sequence are known as its '''terms'''. Let $\sequence {x_n}$ be a sequence. Then the '''$k$th term''' of $\sequence {x_n}$ is the ordered pair $\tuple {k, x_k}$.
Definition:Totally Ordered Group
A '''totally ordered group''' is a totally ordered structure $\left({G, \circ, \preceq}\right)$ such that $\left({G, \circ}\right)$ is a group.
Definition:Jacobson Radical
Let $R$ be a commutative ring with unity. Let $\operatorname{maxspec} \left({R}\right)$ be the set of maximal ideals of $R$. Then the '''Jacobson radical''' of $R$ is: :$\displaystyle \operatorname{Jac} \left({R}\right) = \bigcap_{m \mathop \in \operatorname{maxspec} \left({R}\right)} m$ That is, it is the intersection of all maximal ideals of $R$.
Definition:Jacobson Ring
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity. Then $\left({R, +, \circ}\right)$ is a '''Jacobson ring''' : : every prime ideal of $\left({R, +, \circ}\right)$ is an intersection of  maximal ideals.
Definition:Adjugate Matrix
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $\mathbf C$ be its cofactor matrix. The '''adjugate matrix''' of $\mathbf A$ is the transpose of $\mathbf C$: :$\adj {\mathbf A} = \mathbf C^\intercal$
Definition:Radical of Ideal of Ring
Let $A$ be a commutative ring with unity. Let $I$ be an ideal of $A$. === Definition 1 === {{Definition:Radical of Ideal of Ring/Definition 1}} === Definition 2 === {{Definition:Radical of Ideal of Ring/Definition 2}}
Definition:Radical Ideal of Ring
Let $A$ be a commutative ring with unity. Let $I$ be an ideal of $A$. Then $I$ is a '''radical ideal'''  it is equal to its radical.
Definition:Nilpotent Ring Element
Let $R$ be a ring with zero $0_R$. An element $x \in R$ is '''nilpotent''' : :$\exists n \in \Z_{>0}: x^n = 0_R$
Definition:Reduced Ring
Let $\struct {R, +, \circ}$ be a ring. Then $R$ is a '''reduced ring'''  it contains no nilpotent elements except the zero.
Definition:Nilradical of Ring
Let $A$ be a commutative ring with unity. === Definition 1 === {{:Definition:Nilradical of Ring/Definition 1}} === Definition 2 === {{:Definition:Nilradical of Ring/Definition 2}}
Definition:Polynomial Ring/Monoid Ring on Free Monoid on Set
Let $R \sqbrk {\family {X_i: i \in I} }$ be the ring of polynomial forms in $\family {X_i: i \in I}$. The '''polynomial ring in $I$ indeterminates over $R$''' is the ordered triple $\struct {\struct {A, +, \circ}, \iota, \family {X_i: i \in I} }$ {{DefinitionWanted|define the inclusion and indeterminates in this case}}
Definition:Monomial of Free Commutative Monoid
A '''monomial''' in the indexed set $\family {X_j: j \in J}$ is a possibly infinite product: :$\ds \prod_{j \mathop \in J} X_j^{k_j}$ with integer exponents $k_j \ge 0$ such that $k_j = 0$ for all but finitely many $j$. Let $\mathbf X = \family {X_j}_{j \mathop \in J}$ and for a multiindex $k = \paren {k_j}_{j \mathop \in J}$ over $J$ define: :$\ds \mathbf X^k = \prod_{j \mathop \in J} X_j^{k_j}$ Then a '''monomial''' is an object of the form $\mathbf X^k$, where $k$ is a multiindex.
Definition:Free Commutative Monoid
The '''free commutative monoid''' on an indexed set $X = \family {X_j: j \in J}$ is the set $M$ of all monomials under the standard multiplication. That is, it is the set $M$ of all finite sequences of $X$.
Definition:Multiindex
=== Definition 1 === Let $\ds m = \prod_{j \mathop \in J} X_j^{k_j}$ be a monomial in the indexed set $\set {X_j: j \mathop \in J}$. Such a monomial can be expressed implicitly and more compactly by referring only to the sequence of indices: :$k = \sequence {k_j}_{j \mathop \in J}$ and write $m = \mathbf X^k$ without explicit reference to the indexing set. Such an expression is called a '''multiindex''' (or '''multi-index'''). === Definition 2 === Let $J$ be a set. A '''$J$-multiindex''' is a sequence of natural numbers indexed by $J$: :$\displaystyle k = \sequence {k_j}_{j \mathop \in J}$ with only finitely many of the $k_j$ non-zero. === Definition 3 === A '''multiindex''' is an element of $\Z^J$, the free $\Z$-module on $J$, an abelian group of rank over $\Z$ equal to the cardinality of $J$. === Modulus === {{:Definition:Multiindex/Modulus}}
Definition:Pointwise Addition of Mappings
Let $S$ be a non-empty set. Let $\struct {G, \circ}$ be a commutative semigroup. Let $G^S$ be the set of all mappings from $S$ to $G$. Then '''pointwise addition''' on $G^S$ is the binary operation $\circ: G^S \times G^S \to G^S$ (the $\circ$ is the same as for $G$) defined by: :$\forall f, g \in G^S: \forall s \in S: \map {\paren {f \circ g} } s := \map f s \circ \map g s$ The double use of $\circ$ is justified as $\struct {G^S, \circ}$ inherits all abstract-algebraic properties $\struct {G, \circ}$ might have. This is rigorously formulated and proved on Mappings to Algebraic Structure form Similar Algebraic Structure. === Pointwise Multiplication === Let $\circ$ be used with multiplicative notation. Then the operation defined above is called '''pointwise multiplication''' instead.
Definition:Ring of Polynomial Functions
Let $\struct {R, +, \circ}$ be a commutative ring with unity. Let $R \sqbrk {\set {X_j: j \in J} }$ be the ring of polynomial forms over $R$ in the indeterminates $\set {X_j: j \in J}$. Let $R^J$ be the free module on $J$. Let $A$ be the set of all polynomial functions $R^J \to R$. Then the operations $+$ and $\circ$ on $R$ induce pointwise operations on $A$. We denote these operations by the same symbols: :$\forall x \in R^J: \map {\paren {f + g} } x = \map f x + \map g x$ :$\forall x \in R^J: \map {\paren {f \circ g} } x = \map f x \circ \map g x$ The '''ring of polynomial functions''' is the resulting algebraic structure.
Definition:Arithmetic Function
An '''arithmetic function''' or '''arithmetical function''' is a complex valued function defined on the strictly positive integers. That is, a mapping: :$f: \Z_{>0} \to \C$
Definition:Category
A '''category''' is an interpretation of the metacategory axioms within set theory. Because a metacategory is a metagraph, this means that a '''category''' is a graph. Let $\mathfrak U$ be a class of sets. A metacategory $\mathbf C$ is a '''category''' : :$(1): \quad$ The objects form a subset $\mathbf C_0$ or $\operatorname {ob} \ \CC \subseteq \mathfrak U$ :$(2): \quad$ The morphisms form a subset $\mathbf C_1$ or $\operatorname{mor} \ \mathbf C$ or $\operatorname{Hom} \ \mathbf C \subseteq \mathfrak U$
Definition:Abuse of Notation
Mathematical notation can be considered in two ways: * As an aid to mathematical understanding, no more and no less than a useful convention to encapsulate more or less complicated ideas in a completely unambiguous format; * As the reason for mathematical effort, so as to encapsulate a truth as a documented piece of aesthetic beauty in its own right. '''Abuse of notation''' is a technique of using a system of symbology in a way different from that for which it was originally defined. Such abuse may make a train of thought more streamlined, as it is often possible to save considerable redefinition of one's terms. However, such abuse is frequently considered to be incorrect, improper and (in the eyes of many mathematicians) illegal. Philosophers of the various schools practising pragmatism tend to consider that if a notation has been adequately explained, then one should be allowed to use it in whatever way is most useful to communicate one's ideas. On the other hand, such an attitude causes indignation, rage and fury among philosophers whose attraction to mathematics is purely aesthetic.
Definition:Completion (Metric Space)
Let $M_1 = \struct {A, d}$ and $M_2 = \struct {\tilde A, \tilde d}$ be metric spaces. Then $M_2$ is a '''completion''' of $M_1$, or $M_2$ '''completes''' $M_1$, : :$(1): \quad M_2$ is a complete metric space :$(2): \quad A \subseteq \tilde A$ :$(3): \quad A$ is dense in $M_2$ :$(4): \quad \forall x, y \in A : \map {\tilde d} {x, y} = \map d {x, y}$. In terms of restriction of functions, this says that $\map {\tilde d {\restriction_A} } = d$. It is immediate from this definition that a '''completion''' of a metric space $M_1$ consists of: :A complete metric space $M_2$ :An isometry $\phi : A \to \tilde A$ such that $\map \phi A = \set {\map \phi x: x \in A}$ is dense in $M_2$. An isometry is often required to be bijective, so here one should consider $\phi$ as a mapping from $A$ to the image of $\phi$. Therefore to insist that $\phi$ be an isometry, ''in this context'', is to say that $\phi$ must be an injection that preserves the metric of $M_1$.
Definition:Metacategory
A '''metacategory''' is a metagraph subject to extra restrictions. As such, a '''metacategory''' $\mathbf C$ consists of: * A collection $\mathbf C_0$ of objects $X, Y, Z, \ldots$ * A collection $\mathbf C_1$ of morphisms $f, g, h, \ldots$ between its objects The morphisms of $\mathbf C$ are subjected to: {{begin-axiom}} {{axiom | n = C1         | lc= ''Composition''         | t = For objects $X, Y, Z$ and morphisms $X \stackrel {f} {\longrightarrow} Y \stackrel {g} {\longrightarrow} Z$ with $\operatorname{cod} f = \operatorname{dom} \, g$, there exists a morphism: :$g \circ f : X \to Z$ called the composition of $f$ and $g$. }} {{axiom | n = C2         | lc= ''Identity''          | t = For every object $X$, there is a morphism ${\operatorname{id}_X}: X \to X$, called the identity morphism, such that: :$f \circ {\operatorname{id}_X} = f$, and ${\operatorname{id}_X} \circ g = g$ for any object $Y$ and morphisms $f : X \to Y$ and $g : Y \to X$ }} {{axiom | n = C3         | lc= ''Associativity''         | t = For any three morphisms $f, g, h$: :$f \circ \left({g \circ h}\right) = \left({f \circ g}\right) \circ h$ whenever these compositions are defined (according to $(C1)$). }} {{end-axiom}} To describe a '''metacategory''' it is necessary to specify: :* The collection $\mathbf C_0$ of objects :* The collection $\mathbf C_1$ of morphisms :* For each object $X$, an identity morphism ${\operatorname{id}_X}: X \to X$ :* For every appropriate pair of morphisms $f, g$, the composite morphism $g \circ f$ However, the last two are often taken to be implicit when the objects and morphisms are familiar. Of course, after defining these, it is still to be shown that $(C1)$ up to $(C3)$ are satisfied. A '''metacategory''' is purely axiomatic, and does not use set theory. For example, the objects are not "elements of the set of objects", because these axioms are (without further interpretation) unfounded in set theory. For some purposes, it is convenient to have a different description of a '''metacategory'''. Two such descriptions are found on: * Characterization of Metacategory via Equations * Morphisms-Only Metacategory
Definition:Non-Archimedean
=== Non-Archimedean Norm (Vector Space) === {{:Definition:Non-Archimedean/Norm (Vector Space)}} === Non-Archimedean Norm (Division Ring) === {{:Definition:Non-Archimedean/Norm (Division Ring)}} === Non-Archimedean Metric === {{:Definition:Non-Archimedean/Metric}}
Definition:Orthogonal (Linear Algebra)
Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space. Let $u, v \in V$. Then $u$ and $v$ are '''orthogonal''' : :$\innerprod u v = 0$ === Orthogonal Set === {{:Definition:Orthogonal (Linear Algebra)/Set}} === Vectors in $\R^n$ === {{:Definition:Orthogonal (Linear Algebra)/Real Vector Space}}
Definition:Valued Field
Let $\struct {K, +, \circ}$ be a field. Let $\norm {\,\cdot\,}$ be a norm on $K$. Then $\struct {K, \norm{\,\cdot\,} }$ is a '''valued field'''.
Definition:Minkowski Functional
Let $E$ be a vector space over $\R$. A functional $p: E \to \R$ is called a '''Minkowski functional''' if it satisfies: {{begin-axiom}} {{axiom | n = 1         | q = \forall x \in E, \forall \lambda \in \R_{>0}         | ml= \map p {\lambda x}         | mo= =         | mr= \lambda \map p x         | rc= that is, $p$ is positive homogeneous }} {{axiom | n = 2         | q = \forall x, y \in E         | ml= \map p {x + y}         | mo= \le         | mr= \map p x + \map p y         | rc= that is, $p$ is sub-additive }} {{end-axiom}} {{NamedforDef|Hermann Minkowski|cat = Minkowski}}
Definition:Linear Functional
Let $E$ be a vector space over a field $\Bbb F$. Let $D$ be a linear subspace of $E$. A mapping $f : D \to \Bbb F$ is called a '''linear functional''' iff: :$ f \left({\alpha x + \beta y}\right) = \alpha f(x) + \beta f(y)$ holds for all $x, y$ in $L$ and for all $\alpha, \beta$ in $\Bbb F$.
Definition:Euclidean Domain
Let $\struct {R, +, \circ}$ be an integral domain with zero $0_R$. Let there exist a mapping $\nu: R \setminus \set {0_R} \to \N$ with the properties: :$(1): \quad$ For all $a, b \in R, b \ne 0_R$, there exist $q, r \in R$ with $\map \nu r < \map \nu b$, or $r = 0_R$ such that: ::::$a = q \circ b + r$ :$(2): \quad$ For all $a, b \in R, b \ne 0_R$: ::::$\map \nu a \le \map \nu {a \circ b}$ Then $\nu$ is called a '''Euclidean valuation''' and $R$ is called a '''Euclidean domain'''.
Definition:Farey Sequence
The '''Farey sequence''' is a chain of subsets of the reduced rational numbers lying in $\Q \cap \closedint 0 1$. For $Q \in \Z_{>0}$, the '''Farey set''' $F_Q$ is the set of all reduced rational numbers with denominators not larger than $Q$: :$F_Q = \set {\dfrac p q: p = 0, \ldots, Q,\ q = 1, \ldots, Q,\ p \perp q}$ where $p \perp q$ denotes that $p$ and $q$ are coprime.
Definition:Linear Representation
=== Groups === {{:Definition:Linear Representation/Group}} === Algebras === {{:Definition:Linear Representation/Algebra}}
Definition:Dimension (Representation Theory)
Let $\struct {k, +, \circ}$ be a field. Let $V$ be a vector space over $k$ of finite dimension. Let $\GL V$ be the general linear group of $V$. Let $\struct {G, \cdot}$ be a finite group. Let $\rho: G \to \GL V$ be a linear representation of $G$ on $V$. The '''dimension''' or '''degree''' of $\rho$, written $\map \deg \rho$ is the dimension of the vector space $V$.
Definition:Faithful Linear Representation of Group
Let $ \left({k, +, \circ}\right)$ be a field. Let $V$ be a vector space over $k$ of finite dimension. Let $\operatorname{GL} \left({V}\right)$ be the general linear group of $V$. Let $\left({G, \cdot}\right)$ be a finite group. Let $\rho : G \to \operatorname{GL} \left({V}\right)$ be a linear representation of $G$ on $V$. Then $\rho$ is '''faithful''' if the kernel of $\rho$ is trivial.  2gwgnr1htwebwxkr89qk9ty3psbwpvz
Definition:Linear Group Action
Let $\left({V, +, \cdot}\right)$ be a vector space over a field $\left({k, \oplus, \circ}\right)$. Let $G$ be a group. Let $\phi : G \times V \to V$ be an action of $G$ on $V$. Then $\phi$ is a '''(left) linear group action'''  it is compatible with the linear structure of $V$ in the following sense: : $(1): \quad \forall v_1, v_2 \in V: g \in G: \phi \left({g, v_1 + v_2}\right) = \phi \left({g, v_1}\right) + \phi \left({g, v_2}\right)$ : $(2): \quad \forall \lambda \in k, g \in G, v \in V: \phi \left({g, \lambda \cdot v}\right) = \lambda \cdot \phi \left({g, v}\right)$ === Right Linear Group Action === {{:Definition:Linear Group Action/Right Linear Group Action}}
Definition:Group Algebra
Let $\left({k, + ,\circ}\right)$ be a field. Let $\left({G, *}\right)$ be a finite group. Then the '''group algebra''' $k G$ or $k \left[{G}\right]$ is the set of all formal sums: :$\displaystyle \sum_{g \in G} \alpha_g g\ :\ \alpha_g \in k$ That is, $k \left[{G}\right]$ is the free vector space over $k$ with basis $G$.  nne3ovn9w8raya0d71gehml4l2xsazi
Definition:Krull Dimension of Ring
Let $\struct {R, +, \circ}$ be a commutative ring with unity. The '''Krull dimension''' of $R$ is the supremum of lengths of chains of prime ideals, ordered by inclusion: :$\map {\operatorname {dim_{Krull} } } R = \sup \set {n \in \N: \exists p_0, \ldots, p_n \in \Spec R: \mathfrak p_0 \subsetneqq \mathfrak p_1 \subsetneqq \cdots \subsetneqq \mathfrak p_n}$ where $\Spec R$ is the prime spectrum of $R$. That is, the '''Krull dimension''' is $\infty$ if there exist arbitrarily long chains.
Definition:Exact Sequence of Modules
Let $\left({R, +, \cdot}\right)$ be a ring. Let: :$(1): \quad \cdots \longrightarrow M_i \stackrel{d_i} {\longrightarrow} M_{i + 1} \stackrel {d_{i + 1}} {\longrightarrow} M_{i + 2} \stackrel {d_{i + 2}} {\longrightarrow} \cdots$ be a sequence of $R$-modules $M_i$ and $R$-module homomorphisms $d_i$. Then the sequence $(1)$ is '''exact''' : : $\forall i: \operatorname{Im} d_i = \ker d_{i + 1}$ where $\operatorname{Im}$ and $\ker$ denote the image and kernel of mappings respectively.
Definition:Differential Module
Let $R$ be a commutative ring with unity. Let $M$ be an $R$-module. A '''differential''' on $M$ is a homomorphism $d:M\to M$ of $R$-modules such that $d^2 = d\circ d = 0$. The pair $(M,d)$ is called a '''differential module'''.  3bgo6tem59m2851njehp4d3uddkt2p1
Definition:Graded Ring
Let $M$ be a semigroup. An '''$M$-graded ring''' is a pair $\struct {R, f}$ where: :$R$ is a ring :$f$ is a gradation of type $M$ on the additive group of $R$ which is compatible with the ring structure of $R$.
Definition:Graded Module
Let $R$ be a graded commutative ring with unity. Let $M$ be an $R$-module. We say $M$ is $\N$-'''graded''' if it has a decomposition as a direct sum of abelian groups :$\displaystyle R = \bigoplus_{n \in \N} R_n $ such that for all $m, n \in \N$ we have $R_n M_m \subseteq M_{n+m}$. Similarly, we say $M$ is $\Z$-'''graded''' if it has a decomposition as a direct sum of abelian groups :$\displaystyle R = \bigoplus_{n \in \Z} R_n $ such that for all $m,n \in \Z$ we have $R_nM_m \subseteq M_{n+m}$.  q984qkm4593t5sqg0wbyu6xtemvda2q
Definition:Differential Complex
Let $R$ be a commutative ring with unity. Let $\displaystyle M = \bigoplus_{n \in \Z} M^n$ be a $\Z$-graded $R$-module that is also a differential module with differential $d$. Then $M$ is a '''differential complex''' if the differential $d$ satisfies: :$ d \left({M^n}\right) \subseteq M^{n+1}$ for all $n \in \Z$. The notation $d_n := d \restriction_{M_n}$ is often seen.
Definition:Cohomology Groups
{{refactor|various definitions to separate}} Let $\left({M, d}\right)$ be a differential complex with grading: :$\displaystyle M = \bigoplus_{n \in \Z} M^n$ Let $d_n := d \restriction_{M_n}$. Elements of the module $M$ are called '''cochains'''. Elements of the submodule $Z^n \left({M}\right) = \ker \left({d_n}\right)$ are called '''cocycles'''. Elements of the submodule $B^n \left({M}\right) = \operatorname{Im} \left({d_{n-1}}\right)$ are called '''coboundaries'''. The modules (and hence groups) $H^n \left({M}\right) = Z^n \left({M}\right)/B^n \left({M}\right)$ are called the '''cohomology groups''' of the differential complex $\left({M, d}\right)$.
Definition:Algebraic Number Field
An '''algebraic number field''' is a finite extension of the field of rational numbers $\Q$.
Definition:Algebraic Integer
Let $K / \Q$ be a number field. Then $\alpha \in K$ is an '''algebraic integer''' if it satisfies a monic polynomial $f \in \Z \left[{X}\right]$. The set of all '''algebraic integers''' in $K$ is denoted $\mathfrak o_K$ or $\mathcal O_K$. By Ring of Algebraic Integers it is a ring, hence usually referred to as the '''ring of algebraic integers''' of $K$. === Quadratic Integer === {{:Definition:Algebraic Integer/Quadratic}}
Definition:Field Norm
Let $K$ be a field and $L / K$ a finite field extension of $K$. Let $\alpha\in L$. === Definition 1 === {{Definition:Field Norm/Definition 1}} === Definition 2: for Galois extensions === {{Definition:Field Norm/Definition 2}}
Definition:Trace (Field Theory)
Let $K$ be a field and $L / K$ a finite field extension of $K$. Then by Vector Space on Field Extension is Vector Space, $L$ is naturally a vector space over $K$. Let $\alpha \in L$, and $\theta_\alpha$ be the linear operator: :$\theta_\alpha: L \to L: \beta \mapsto \alpha\beta$ The '''trace''' $\map {\operatorname {Tr}_{L / K} } \alpha$ of $\alpha$ is the trace of this linear operator.
Definition:Content of Polynomial
=== Integer Polynomial === {{:Definition:Content of Polynomial/Integer}} === Rational Polynomial === {{:Definition:Content of Polynomial/Rational}} === Polynomial in GCD Domain === {{:Definition:Content of Polynomial/GCD Domain}} == Commutative Ring with Unity == {{:Definition:Content of Polynomial/Commutative and Unitary Ring}}
Definition:Ordered Integral Domain
=== Definition 1 === {{:Definition:Ordered Integral Domain/Definition 1}} === Definition 2 === {{:Definition:Ordered Integral Domain/Definition 2}} An ordered integral domain can be denoted: :$\struct {D, +, \times \le}$ where $\le$ is the total ordering induced by the strict positivity property.
Definition:Strict Positivity Property
{{begin-axiom}} {{axiom | n = \text P 1         | lc= Closure under Ring Addition:         | q = \forall a, b \in D         | m = \map P a \land \map P b \implies \map P {a + b} }} {{axiom | n = \text P 2         | lc= Closure under Ring Product:         | q = \forall a, b \in D         | m = \map P a \land \map P b \implies \map P {a \times b} }} {{axiom | n = \text P 3         | lc= Trichotomy Law:         | q = \forall a \in D         | m = \paren {\map P a} \lor \paren {\map P {-a} } \lor \paren {a = 0_D} }} {{axiom | lc= For $\text P 3$, '''exactly one''' condition applies for all $a \in D$. }} {{end-axiom}}
Definition:Strict Negativity Property
Let $\struct {D, +, \times}$ be an ordered integral domain, whose (strict) positivity property is denoted $P$. The '''strict negativity property''' $N$ is defined as: :$\forall a \in D: \map N a \iff \map P {-a}$ This is compatible with the trichotomy law: :$\forall a \in D: \map P a \lor \map P {-a} \lor a = 0_D$ which can therefore be rewritten: :$\forall a \in D: \map P a \lor \map N a \lor a = 0_D$ or even: :$\forall a \in D: \map N a \lor \map N {-a} \lor a = 0_D$