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

Every plane that could fly was sent into the air.

Every hyperbola is congruent to the origin-centered East-West opening hyperbola sharing its same eccentricity ε ( its shape, or degree of " spread "), and is also congruent to the origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it can be rotated so that it opens in the desired direction and can be translated ( rigidly moved in the plane ) so that it is centered at the origin.

Every atom across this plane has an individual set of emission cones .</ p > < p > Drawing the billions of overlapping cones is impossible, so this is a simplified diagram showing the extents of all the emission cones combined.

Every vibrating object tends to maintain its plane of vibration if its support is rotated, a result of Newton's first law.

Every immersed curve in the plane admits two possible orientations.

Every outerplanar graph can be represented as an intersection graph of axis-aligned rectangles in the plane, so outerplanar graphs have boxicity at most two.

Every improper rotation of three-dimensional Euclidean space is rotation followed by a reflection in a plane through the origin.

Every smooth surface S has a unique affine plane tangent to it at each point.

* Every triangle group T is a discrete subgroup of the isometry group of the sphere ( when T is finite ), the Euclidean plane ( when T has a Z + Z subgroup of finite index ), or the hyperbolic plane.

Every minimal projective ruled surface other than the projective plane is the projective bundle of a 2-dimensional vector bundle over some curve.

Every plane travelling to and from Europe or North America must talk to either or both of these air traffic controls ( ATC ).

Every planar graph has a flat and linkless embedding: simply embed the graph into a plane and embed the plane into space.

Every detail of these patterns acts as a consistent portal to a different kingdom inside the plane, which itself comprises many separate realms.

Every point in the plane has at least one tangent line to γ passing through it, and so region filled by the tangent lines is the whole plane.

Every detail of these patterns acts as a consistent portal to a different kingdom inside the plane, which itself comprises many separate realms.

Every set of points in the plane, and the lines connecting them, may be abstracted as the elements and flats of a rank-3 oriented matroid.

Every second-order linear ODE on the extended complex plane with at most four regular singular points, such as the Lamé equation or the hypergeometric differential equation, can be transformed into this equation by a change of variable.

Every plane B that is completely orthogonalTwo flat subspaces S < sub > 1 </ sub > and S < sub > 2 </ sub > of dimensions M and N of a Euclidean space S of at least M + N dimensions are called completely orthogonal if every line in S1 is orthogonal to every line in S2.

* Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, every degree n Bézier curve is also a degree m curve for any m > n. In detail, a degree n curve with control points P < sub > 0 </ sub >, …, P < sub > n </ sub > is equivalent ( including the parametrization ) to the degree n + 1 curve with control points P '< sub > 0 </ sub >, …, P '< sub > n + 1 </ sub >, where.

Every space filling curve hits some points multiple times, and does not have a continuous inverse.

Every point on the Lorenz curve represents a statement like " the bottom 20 % of all households have 10 % of the total income.

Every study has found that instead of people's MBTI scores clustering around two opposite poles, such as intuition vs. sensation, with few people scoring in the middle, people's scores actually cluster around the middle of each scale in a bell curve.

Every Edwards curve with a point of order 3 can be written in the ways shown above.

Every closed curve c on X is homologous to for some simple closed curves c < sub > i </ sub >, that is,

Every irreducible non-degenerate curve of degree is a rational normal curve.