If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

Some authors also require the domain of the Euclidean function be the entire ring R ; this can always be accommodated by adding 1 to the values at all nonzero elements, and defining the function to be 0 at the zero element of R, but the result is somewhat awkward in the case of K. The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set ; this weakening does not affect the most important implications of the Euclidean property.

* K < nowiki ></ nowiki > X < nowiki ></ nowiki >, the ring of formal power series over the field K. For each nonzero power series P, define f ( P ) as the degree of the smallest power of X occurring in P. In particular, for two nonzero power series P and Q, f ( P )≤ f ( Q ) iff P divides Q.

The converse does not hold since for any field K, K is a UFD but is not a PID ( to prove this look at the ideal generated by It is not the whole ring since it contains no polynomials of degree 0, but it cannot be generated by any one single element ).

All members of the vitamin K group of vitamins share a methylated naphthoquinone ring structure ( menadione ), and vary in the aliphatic side chain attached at the 3-position ( see figure 1 ).

* The maximal ideals of the polynomial ring over an algebraically closed field K are the ideals of the form.

When specializing to the preadditive categories of abelian groups or modules over a ring, this notion of kernel coincides with the ordinary notion of a kernel of a homomorphism, if one identifies the ordinary kernel K of f: A → B with its embedding K → A.

* K: rings of polynomials in one variable with coefficients in a field.

Let K be R, C, or any field, and let V be the set P of all polynomials with coefficients taken from the field K.

An immediate corollary is the " weak Nullstellensatz ": The ideal I in kX < sub > n </ sub > contains 1 if and only if the polynomials in I do not have any common zeros in K < sup > n </ sup >.

An extension L which is a splitting field for multiple polynomials p ( X ) over K is called a normal extension.

Such a Galois closure should contain a splitting field for all the polynomials p over K that are minimal polynomials over K of elements a of K ′.

If a is algebraic over K, then there are many non-zero polynomials g ( x ) with coefficients in K such that g ( a ) = 0.

* the commutative algebra K of all polynomials over K.

* The square n-by-n matrices with entries from the field K form a unitary associative algebra over K.

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.

because if M is any algebraically closed field containing K, then the elements of M which are algebraic over K form an algebraic closure of K.

For example, if K is a field of characteristic p and if X is transcendental over K, is a non-separable algebraic field extension.

Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.

The class of the congruence is Af, since an arbitrary plane meets **zg in K points.

Since the complex of singular lines is of order K and since there is no complex of invariant lines, it follows from the formula Af that the order of the involution is Af.

Since C is rational, this correspondence has K coincidences, each of which implies a line of the pencil which meets its image.

For if such were the case, either the plane of the two lines would meet **zg in more than K points or, alternatively, the order of the image regulus of the pencil determined by the two lines would be too high.

The K factor, a term used to denote the rate of heat transmission through a material ( B.t.u./sq. ft. of material/hr./*0F./in. of thickness ) ranges from 0.24 to 0.28 for flexible urethane foams and from 0.12 to 0.16 for rigid urethane foams, depending upon the formulation, density, cell size, and nature of blowing agents used.

Table 1,, p. 394, shows a comparison of K factor ratings of a number of commercial insulating materials in common use, including two different types of rigid urethane foam.

The man most firmly at grips with the problem is the University of Minnesota's Physiologist Ancel Keys, 57, inventor of the wartime K ( for Keys ) ration and author of last year's bestselling Eat Well And Stay Well.

* in J. Barnes, M. Schofield, and R. R. K. Sorabji, eds .( 1975 ).

* Pfeiff, K. A., 1943.

The equilibrium constant K is an expression of the equilibrium concentrations of the molecules or the ions in solution.

The acid dissociation constant K < sub > a </ sub > is generally used in the context of acid-base reactions.

The stronger of two acids will have a higher K < sub > a </ sub > than the weaker acid ; the ratio of hydrogen ions to acid will be higher for the stronger acid as the stronger acid has a greater tendency to lose its proton.

Charles K. Smith argues that Swift ’ s rhetorical style persuades the reader to detest the speaker and pity the Irish.

To do this, Moseley measured the wavelengths of the innermost photon transitions ( K and L lines ) produced by the elements from aluminum ( Z = 13 ) to gold ( Z = 79 ) used as a series of movable anodic targets inside an x-ray tube.

Maj. Gen. Polk ignored the problems of Fort Henry and Fort Donelson when he took command and, after Johnston took command, at first refused to comply with Johnston's order to send an engineer, Lt. Joseph K. Dixon, to inspect the forts.

For, the letters associated with those numbers are K, L, M, N, O, ..., respectively.

For the specific heats at least, the limiting value itself is definitely zero, as borne out by experiments to below 10 K. Even the less detailed Einstein model shows this curious drop in specific heats.

For example the United States uses NTSC-M, the UK uses PAL-I, France uses SECAM-L, much of Western Europe and Australia uses PAL-B / G, most of Eastern Europe uses PAL-D / K or SECAM-D / K and so on.

For a time, memory capacities were often expressed in K, even when M could have been used: The IBM System / 370 Model 158 brochure ( 1972 ) had the following: " Real storage capacity is available in 512K increments ranging from 512K to 2, 048K bytes.

For each K, the function E < sub > K </ sub >( P ) is required to be an invertible mapping on

For example, trace ( but detectable ) amounts of carbon-14 (< sup > 14 </ sup > C ) are continually produced in the atmosphere by cosmic rays impacting nitrogen atoms, and argon-40 (< sup > 40 </ sup > Ar ) is continually produced by the decay of primordially occurring but unstable potassium-40 (< sup > 40 </ sup > K ).

For the Sun and stars with low temperatures, the prominence of the H and K lines can be an indication of strong magnetic activity in the chromosphere.

For example, Philip K. Dick's works contain recurring themes of social decay, artificial intelligence, paranoia, and blurred lines between objective and subjective realities, and the influential cyberpunk movie Blade Runner is based on one of his books.

A fan of Philip K. Dick, author of " We Can Remember it For You Wholesale ," the short story upon which the film was based, Cronenberg related ( in the biography / overview of his work, Cronenberg on Cronenberg ) that his dissatisfaction with what he envisioned the film to be and what it ended up being pained him so greatly that for a time, he suffered a migraine just thinking about it, akin to a needle piercing his eye.

For instance, Freeze Arrows ( introduced in DDRMAX ) which is a long green arrow that must be held down until the tail of it reaches the Step Zone, that is given an " O. K .!

For example, the K ' iche ' language spoken in Guatemala has the inflectional prefixes k-and x-to mark incompletive and completive aspect ; Mandarin Chinese has the aspect markers-le 了 ,-zhe 着, zài-在, and-guò 过 to mark the perfective, durative stative, durative progressive, and experiential aspects, and also marks aspect with adverbs ; and English marks the continuous aspect with the verb to be coupled with present participle and the perfect with the verb to have coupled with past participle.

For example, Q-Q-K-3-2 would win ( because its K kicker outranks the 10 ), but Q-Q-10-4-3 would lose ( because its 4 is outranked by the 5 ).

For example, assume that a Texas hold ' em board looks like this after the third round: 5 ♠ < font color = red > K ♦ 7 ♦</ font > J ♠, and that a player is holding < font color = red > A ♦ 10 ♦</ font >.

For example, in the Harry Potter novels, J. K. Rowling reshapes a standard English proverb into “ It ’ s no good crying over spilt potion ” and Dumbledore

For example, with K ♠- 9 ♠- 8 ♣- 7 ♠- 6 ♣- 5 ♠- 4 ♠ playing the flush would put 8-6 in front, playing the 9-high straight would put K-4 up front, but the correct play is K-9 and 8-7-6-5-4.

For the limit process, we use the restriction homomorphisms Gal ( F < sub > 1 </ sub >/ K ) → Gal ( F < sub > 2 </ sub >/ K ), where F < sub > 2 </ sub > ⊆ F < sub > 1 </ sub >.