** 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 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 algebraic extension of k is separable.

* 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 lattice element of the structure is in its proper place, whether it is a single atom or a molecular grouping.

Every finite tree structure has a member that has no superior.

Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes ( or congruence classes ) for the relation.

Every group has a presentation, and in fact many different presentations ; a presentation is often the most compact way of describing the structure of the group.

Every solid has its own characteristic energy band structure.

Every chapter contains a comprehensive compilation all of the published examples of the reaction organized in tables according to the structure of the starting material.

Every Riemann surface is a two-dimensional real analytic manifold ( i. e., a surface ), but it contains more structure ( specifically a complex structure ) which is needed for the unambiguous definition of holomorphic functions.

: Every oriented prime closed 3-manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.

* " What Every Programmer Should Know About Memory " by Ulrich Drepper — explains the structure of modern memory subsystems and suggests how to utilize them efficiently

Every H * is very special in structure: it is pure-injective ( also called algebraically compact ), which says more or less that solving equations in H * is relatively straightforward.

Every structure is associated with a certain quantity of energy, which determines the stability of the molecule or ion ( the lower energy, the greater stability ).

Every Hermitian manifold is a complex manifold which comes naturally equipped with a Hermitian form and an integrable, almost complex structure.

Every animal cell is enclosed in a plasma membrane, which has the structure of a lipid bilayer with many types of large molecules embedded in it.

Every structure from the World's Columbian Exposition was long ago destroyed by fire, demolished or moved elsewhere, except for the old Palace of Fine Arts, now the Museum of Science and Industry, The Palace of Fine Arts, the only fireproof building at the fair, fell into disrepair and was rehabilitated with a $ 5 million grant in 1930 from Julius Rosenwald ( President of Sears, Roebuck and Co .).

Every five years, starting the day after Labor Day, the New Jersey Department of Environmental Protection ’ s ( DEP ) water level management plan allows the lake to be lowered five feet to allow for inspection of the dam structure, property owners ’ repairs to lakeshore structures, aquatic vegetation control, and silt and drainage material removal from areas around the lake.

Every timber structure on the site had been burnt, the charcoal being the only organic matter that survived the acid soils.

Every server uses its own query logic and structure.

Every ( normal ) Boolean algebra with operators can be represented as a field of sets on a relational structure in the sense that it is isomorphic to the complex algebra corresponding to the field.

Every compact smooth manifold of dimension 2n, which has only handles of index ≤ n, has a Stein structure provided n > 2, and when n = 2 the same holds provided the 2-handles are attached with certain framings ( framing less than the Thurston-Bennequin framing ).

Every window was lit, to lurid effect, as if the interior of the structure might be on fire.

Every Menards store shares a common structure.

Every structure shown here was built during this period, along with tens of others