** Every field has an algebraic closure.

Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic.

Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.

Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories.

* Every root of a monic polynomial whose coefficients are algebraic integers is itself an algebraic integer.

Every algebraic structure has its own notion of homomorphism, namely any function compatible with the operation ( s ) defining the structure.

Every algebraic curve C of genus g ≥ 1 is associated with an abelian variety J of dimension g, by means of an analytic map of C into J.

* Every nonempty affine algebraic set may be written uniquely as a union of algebraic varieties ( where none of the sets in the decomposition are subsets of each other ).

Every field has an algebraic closure, and it is unique up to an isomorphism that fixes F.

* Every holomorphic vector bundle on a projective variety is induced by a unique algebraic vector bundle.

* Every ( biregular ) algebraic automorphism of a projective space is projective linear.

* Every real algebraic number field K of degree n contains a PV number of degree n. This number is a field generator.

* Every substructure is the union of its finitely generated substructures ; hence Sub ( A ) is an algebraic lattice.

Also, a kind of converse holds: Every algebraic lattice is isomorphic to Sub ( A ) for some algebra A.

* Every character value is a sum of n m < sup > th </ sup > roots of unity, where n is the degree ( that is, the dimension of the associated vector space ) of the representation with character χ and m is the order of g. In particular, when F is the field of complex numbers, every such character value is an algebraic integer.

* Every finite poset is directed complete and algebraic.

* Let K < sup > a </ sup > be an algebraic closure of K containing L. Every embedding σ of L in K < sup > a </ sup > which restricts to the identity on K, satisfies σ ( L ) = L. In other words, σ is an automorphism of L over K.

Every algebraic plane curve has a degree, which can be defined, in case of an algebraically closed field, as number of intersections of the curve with a generic line.

Every planar graph has an algebraic dual, which is in general not unique ( any dual defined by a plane embedding will do ).

** Every field extension has a transcendence basis.

* Every finite extension of k is separable.

* Every reduced commutative k-algebra A is a separable algebra ; i. e., is reduced for every field extension F / k.

Every connected topological group has a unique universal cover as a topological space, which has a unique group structure as a central extension by the fundamental group.

" Every temple or palace -- and by extension, every sacred city or royal residence -- is a Sacred Mountain, thus becoming a Centre.

In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes the matrix algebra by extending scalars (= tensoring with a field extension ), i. e. for a suitable field extension K of F, is isomorphic to the 2 × 2 matrix algebra over K.

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

Every contraction mapping is Lipschitz continuous and hence uniformly continuous ( for a Lipschitz continuous function, the constant k is no longer necessarily less than 1 ).

* Every pair of congruence relations for an unknown integer x, of the form x ≡ k ( mod a ) and x ≡ l ( mod b ), has a solution, as stated by the Chinese remainder theorem ; in fact the solutions are described by a single congruence relation modulo ab.

# Moral law of karma: Every action ( by way of body, speech, and mind ) will have karmic results ( a. k. a. reaction ).

* k = 1 ( criticality ): Every fission causes an average of one more fission, leading to a fission ( and power ) level that is constant.

* Every linear combination of its components Y = a < sub > 1 </ sub > X < sub > 1 </ sub > + … + a < sub > k </ sub > X < sub > k </ sub > is normally distributed.

Every known Sierpinski number k has a small covering set, a finite set of primes with at least one dividing k · 2 < sup > n </ sup >+ 1 for each n > 0.

* Every irreducible closed subset of P < sup > n </ sup >( k ) of codimension one is a hypersurface ; i. e., the zero set of some homogeneous polynomial.

Every maximal outerplanar graph satisfies a stronger condition than Hamiltonicity: it is node pancyclic, meaning that for every vertex v and every k in the range from three to the number of vertices in the graph, there is a length-k cycle containing v. A cycle of this length may be found by repeatedly removing a triangle that is connected to the rest of the graph by a single edge, such that the removed vertex is not v, until the outer face of the remaining graph has length k.

* Every subextension of F / k is separable.

* Every finite subextension of F / k is separable.

Note: Every permutation over a set with k elements is a cyclic permutation of definition type 2 if and only if it is a cyclic permutation of definition type 1 with gcd ( k, offset )

Every vector a in three dimensions is a linear combination of the standard basis vectors i, j, and k.

* Every irreducible polynomial over k has distinct roots.

* Every polynomial over k is separable.

Every degree of freedom in the energy is quadratic and, thus, should contribute k < sub > B </ sub > T to the total average energy, and k < sub > B </ sub > to the heat capacity.

Every k-tree is uniquely ( k + 1 )- colorable.