Every rational number has a unique representation as an irreducible fraction.

Every polynomial in can be factorized into polynomials that are irreducible over F. This factorization is unique up to permutation of the factors and the multiplication of the factors by nonzero constants from F ( because the ring of polynomials over a field is a unique factorization domain whose units are the nonzero constant polynomials ).

Every Heyting algebra with exactly one coatom is subdirectly irreducible, whence every Heyting algebra can be made an SI by adjoining a new top.

* 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 irreducible polynomial in K which has a root in L factors into linear factors in L.

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

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

* Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.

Every polynomial P in x corresponds to a function, ƒ ( x )

Every output of an encoder can be described by its own transfer function, which is closely related to the generator polynomial.

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

Every delta operator ' has a unique sequence of " basic polynomials ", a polynomial sequence defined by three conditions:

Every real polynomial of odd degree has at least one real number as a root.

* Every Jacobi-like polynomial sequence can have its domain shifted and / or scaled so that its interval of orthogonality is, and has Q

* Every Laguerre-like polynomial sequence can have its domain shifted, scaled, and / or reflected so that its interval of orthogonality is, and has Q =

* Every Hermite-like polynomial sequence can have its domain shifted and / or scaled so that its interval of orthogonality is, and has Q

Every polynomial function is a rational function with.

* Every polynomial over k is separable.

Every field and every polynomial ring over a field ( in arbitrarily many variables ) is a reduced ring.

In mathematics, an integer-valued polynomial ( also known as a numerical polynomial ) P ( t ) is a polynomial whose value P ( n ) is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true.

The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.

Every six months the presidency rotates between the states, in an order predefined by the Council's members, allowing each state to preside over the body.

Every year many teams from all over Asia converge in Tokyo to compete.

Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.

Every division ring is therefore a division algebra over its center.

Every field is, of course, one-dimensional over its center.

* Every June, Ellensburg hosts Dachshunds on Parade, an event that draws Dachshund dog owners from all over the Northwest.

Every year sidecar racers travel from all over the UK to Knockhill in Scotland to race in the prestigious ' Jock Taylor Memorial Trophy ' race.

Every day, a blessing is recited over the Lulav and the Etrog.

Every major commercial transport and almost every military jet built in the United States over the last 40 years has been tested in this facility.

Every car uses either a 5. 0 L Ford " Boss 302 " SVO or a 5. 0 L Chevrolet small block race-engine ( depending on the make )-capable of producing between 460 and 485 kW ( 620 — 650 bhp ) of power, but generally quoted as a little over 450 kW ( 600 bhp ) in race trim.

* Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian.

Every day all over the country, police officers are testifying.

Every character in the novel — both men and women — knows their place, and the traditional stereotypes of gender roles are repeated over and over again.

Every Turing machine computes a certain fixed partial computable function from the input strings over its alphabet.

* Every ninth year, people from all over Sweden assembled at the Temple at Uppsala.

Every year, millions of pilgrims from all over the world come to Częstochowa to see it.

Every Calan Mai, the two would fight over Creiddylad, until a battle on Judgement Day, in which the victor would keep her forever.

Controversy ensued over the use of the phrase " Every seat lost to the government is a seat sold to the Boers " as the Unionists waged a personalised campaign against Liberal critics of the war – some posters even portrayed Liberal MPs praising President Kruger and helping him to haul down the Union Jack.

Every year, the post office also creates a special Christmas hand-cancellation pictorial postmark for use during December, which also attracts mail from all over the world.

A quotation attributed to Kolmogorov is into English: " Every mathematician believes that he is ahead over all others.

Every August the Mountain Mama Road Bike Challenge brings cyclists from all over the country to challenge themselves on the steep roads of Highland County.

The argument over the quality of Fitzgerald ’ s translation of the Rubaiyat has, according to Dougan, diverted attention from a fuller understanding of the deeply esoteric message contained in Omar ’ s actual material – " Every line of the Rubaiyat has more meaning than almost anything you could read in Sufi literature ".