In abstract algebra, a field extension L / K is called algebraic if every element of L is algebraic over K, i. e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i. e. which contain transcendental elements, are called transcendental.

Let A be a complex unital Banach algebra in which every non-zero element x is invertible ( a division algebra ).

Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i. e., an element x with.

A product of more than one non-zero rings always has zero divisors: if x is an element of the product all of whose coordinates are zero except p < sub > i </ sub >( x ), and y is an element of the product with all coordinates zero except p < sub > j </ sub >( y ) ( with i ≠ j ), then xy

Given an element φ of H *, the orthogonal complement of the kernel of φ is a one-dimensional subspace of H. Take a non-zero element z in that subspace, and set.

Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory.

Then, for any non-zero element x of M, the cyclic submodule xR must equal M. Fix such an x.

** Unique factorization domain, an integral domain in which every non-zero element can be written as a product of irreducible elements in essentially a unique way

A non-zero element of a ring that is not a zero divisor is called regular.

" More precisely, a member of the Jacobson radical must project under the canonical homomorphism to the zero of every " right division ring " ( each non-zero element of which has a right inverse ) internal to the ring in question.

If R is an ordered field, we can order R (( G )) by setting any element to have the same sign as its leading coefficient, defined as the least element of the index set I associated to a non-zero coefficient.

In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.

( A non-zero non-unit element in a commutative ring is called prime if whenever for some and in, then or.

A field is a commutative ring where every non-zero element a is invertible ; i. e., has a multiplicative inverse b such that a ⋅ b = 1.

Another particular type of element is the zero divisors, i. e. a non-zero element a such that there exists a non-zero element b of the ring such that ab = 0.

Using the standard construction of projective spaces via homogeneous coordinates, the seven points of the Fano plane may be labeled with the seven non-zero ordered triples of binary digits 001, 010, 011, 100, 101, 110, and 111.

disordered and non-zero in the ordered phase.

In a partially ordered set with least element 0, an atom is an element that covers 0, i. e. an element that is minimal among the non-zero elements.

A partially ordered set with a least element is called atomic if every non-zero element b > 0 has an atom a below it, i. e. b ≥ a :> 0.

So here the number operator is normal ordered in the usual sense used in the rest of the article yet its thermal expectation values are non-zero.

The property of having the same orientation defines an equivalence relation on the set of all ordered bases for V. If V is non-zero, there are precisely two equivalence classes determined by this relation.

This yields a formal definition of the field of Puiseux series: it is the direct limit of the direct system, indexed over the non-zero natural numbers n ordered by divisibility, whose objects are all ( the field of formal Laurent series, which we rewrite as

If the base field K is ordered, then the field of Puiseux series over K is also naturally (“ lexicographically ”) ordered as follows: a non-zero Puiseux series f with 0 is declared positive whenever its valuation coefficient is so.

where is the previously defined valuation ( is the first non-zero coefficient ) and ω is infinitely large ( in other words, the value group of is ordered lexicographically, where Γ is the value group of w ).

If only the electric field () is non-zero, and is constant in time, the field is said to be an electrostatic field.

Similarly, if only the magnetic field () is non-zero and is constant in time, the field is said to be a magnetostatic field.

If the pump waves and the signal wave are superimposed in a medium with a non-zero χ < sup >( 3 )</ sup >, this produces a nonlinear polarization field:

A preordered field is a field equipped with a preordering P. The non-zero elements P < sup >*</ sup > form a subgroup of the multiplicative group of F.

A more fundamental restriction is that universal algebra cannot study the class of fields, because there is no type in which all field laws can be written as equations ( inverses of elements are defined for all non-zero elements in a field, so inversion cannot simply be added to the type ).

IEEE 754 NaNs are represented with the exponential field filled with ones ( like infinity values ), and some non-zero number in the significand ( to make them distinct from infinity values ); this representation allows the definition of multiple distinct NaN values, depending on which bits are set in the significand, but also on the value of the leading sign bit ( not all applications are required to provide distinct semantics for those distinct NaN values ).

Binary format NaNs are represented with the exponential field filled with ones ( like infinity values ), and some non-zero number in the significand ( to make them distinct from infinity values ).

If R = K is a field, then a series is invertible if and only if the constant term is non-zero, i. e., if and only if the series is not divisible by X.

In particular, if K is a field then the units of KX < sup >− 1 </ sup > have the form aX < sup > k </ sup >, where a is a non-zero element of K.

However, if R is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of R. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal ( except R ) is a product of prime ideals.

* Applying the divergence theorem to the cross-product of a vector field F and a non-zero constant vector, the following theorem can be proven:

In contrast, for the classical Klein – Gordon field at non-zero temperature, the Gibbs probability density that we would observe a configuration at a time is

However, according to the conjectured gauge-gravity duality ( also known as the AdS / CFT correspondence ), black holes in certain cases ( and perhaps in general ) are equivalent to solutions of quantum field theory at a non-zero temperature.

If the address field is non-zero, it is a disk address of the block, which has previously been rolled out, so the block is fetched from disk and the pbit is set to 1 and the physical memory address updated to point to the block in memory ( another pbit ).

In mathematics, if L is a field extension of K, then an element a of L is called an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial g ( x ) with coefficients in K such that g ( a )= 0.

is non-zero: that is, if ( AC-B < sup > 2 </ sup >/ 4 ) F + BED / 4-CD < sup > 2 </ sup >/ 4-AE < sup > 2 </ sup >/ 4 ≠ 0.

Let φ ( ξ, η, ζ ) be an arbitrary function of three independent variables, and let the spherical wave form F be a delta-function: that is, let F be a weak limit of continuous functions whose integral is unity, but whose support ( the region where the function is non-zero ) shrinks to the origin.

For functions of more than one variable, the theorem states that if the total derivative of a continuously differentiable function F defined from an open set U of R < sup > n </ sup > into R < sup > n </ sup > is invertible at a point p ( i. e., the Jacobian determinant of F at p is non-zero ), then F is an invertible function near p. That is, an inverse function to F exists in some neighborhood of F ( p ).

The non-zero elements of a field F form an abelian group under multiplication ; this group is typically denoted by F < sup >×</ sup >;

; Algebraic extension: If an element α of an extension field E over F is the root of a non-zero polynomial in F, then α is algebraic over F. If every element of E is algebraic over F, then E / F is an algebraic extension.

; Algebraically independent elements: Elements of an extension field of F are algebraically independent over F if they don't satisfy any non-zero polynomial equation with coefficients in F.

The differential operator is called elliptic if the element of Hom ( E < sub > x </ sub >, F < sub > x </ sub >) is invertible for all non-zero cotangent vectors at any point x of X.

More generally, if an irreducible ( non-zero ) polynomial f in F does not have distinct roots, not only must the characteristic of F be a ( non-zero ) prime number p, but also f ( X )= g ( X < sup > p </ sup >) for some irreducible polynomial g in F. By repeated application of this property, it follows that in fact, for a non-negative integer n and some separable irreducible polynomial g in F ( where F is assumed to have prime characteristic p ).