Number of polynomials from the number of coefficients. Kіntsev fields, founded on kіltsy richly segmented. Rich segments in one zminnoy above the field

The ring of rich terms over the field (on the basis of the rich terms over the ring) may have a number of specific powers close to the powers of the ring of integers Z . Podіlnіst richom. Dobre vіdomy for polynomials over the field R in a way similar to "kutom" only arithmetic dії over the coefficients and that's why it's very richly termed over any field k. Vin gives the possibility for two non-zero rich terms p,sk [x] to induce such rich terms q (not more private) and r (surplus), such that p = q * s + r, moreover, either r = 0, or deg (r)< deg(s). Если r =0 , то говорят, что s делит p (или является делителем p) и обозначают это так: s | p. Будем называть многочлен унитарным (или приведенным), если его старший коэффициент равен 1. Определение. Общим наибольшим делителем ненулевых многочленов p и s называется такой унитарный многочлен ОНД(p, s), что 1. ОНД(p, s) | p; ОНД(p, s) | s. 2. q | p, q | s q | ОНД(p, s). По определению, для ненулевого многочлена р со старшим коэффициентом а ОНД (р, 0) = ОНД (0, р) = р/а; ОНД (0, 0)=0. Аналогично определяется ОНД любого числа многочленов. Единственность ОНД двух многочленов непосредственно вытекает из определения. Существование его следует из следующего утверждения. Основная теорема теории делимости (для многочленов). Для любых двух ненулевых многочленов p и q над полем k можно найти такие многочлены u и v над тем же полем, что ОНД(p, q)= u*p+v*q. Доказательство этой теоремы очень похоже на приведенное в лекции доказательство аналогичной теоремы над Z. Все же наметим основные его шаги. Выберем такие многочлены u и v чтобы сумма w= u*p+v*q имела возможно меньшую степень(но была ненулевой!). Можно при этом считать w унитарным многочленом. Проверим, что w | p. Выполняя деление с остатком, получаем: p= s*w+r. Подставляя это равенство в исходное, находим: r = p - s*w =p - s*(u*p+v*q) = (1-s*u)*p+(-s*v)q = U*p + V*q . Если при этом r 0, то deg(r) Замечание. Используя индукцию, можно доказать, что для любого числа многочленов ОНД для подходящих многочленов. Более того, эта формула сохраняется даже для бесконечного множества многочленов, поскольку их ОНД в действительности является ОНД некоторого их конечного подмножества.

Consequence. Be-yaky іdeal іѕ kіltї richly articulated over the field є osnovnym. True, let's say p - OND of all polynomials, which are included in the ideal I. Todi, de For the purpose of the ideal, the stars are clear, which, then, I = (p). Multiplication. Bring the field to deyka, p, q, s are richly articulated over k. If p=q*s, moreover, polynomials q and s can be smaller, lower p, then the polynomial p is called a drive polynomial (over the field k). In the other case, p is irreducible. Irreducible rich term in the ring k[x] is an analogue of the prime number in the ring Z . Understandably, a non-zero polynomial p= can be expanded on TV: p= *, where all polynomials are not reduced over k and may have a higher coefficient equal to 1. It can be brought up to the same exact order of multipliers. Zrozumilo, among the multiples can be the same; such multipliers are called multiples. Combining multiple factors, you can write the same layout like this: p = 0. Apply. one. . It is worth respecting that the rich terms of the first step are not reduced above any field. Multiplier x є multiple, інші - simple. 2. The rich term should not be placed over the field Q of rational numbers. True, if ()=(x-a)*q, then substantiating the equanimity of x=a, it is necessary: , which is impossible for any rational number a. The same polynomial over the field R of speech numbers is induced: , moreover, the other multiplier has a negative discriminant and cannot be expanded over R . Zreshtoyu, over field C complex numbers maєmo: , de = - cubic root of 1. In which case it is possible to understand the reducibility of the source, so that a polynomial can be seen over such a field. The power of rich members, so as not to be induced. 1. Since it is a p-irreducible rich term i d = OND (p, q) 1, then p | q. True, p = d*s і so deg(s)>0, to supercalculate the irreducibility of p, і so deg(s)=0, then d | QP | q. 2. How p | i p is not inducible, but p | chi p | . True, otherwise gcd(p,) = gcd(p,) =1 and therefore the main theorems of the theory of falsity, stars: also, then gcd(p,)=1 i, also, deg(p)=0 .

3. A circle of polynomials over the domain of wholeness.

Further, we will consider only rich terms with coefficients in the dimension of the quantity K (a circle without a zero extension is called the region of quantity), that is. from kіltsya K, for which the addition of two elements can reach zero, for example, one of the multipliers can reach zero. Tse zavzhdi will be swearing at the uvazi, navit yakscho will not be discussed specially.

When adding a richly defined stage n, that stage m is the senior member, as it is in formula (2), more important (the coefficient at ). Since there are no zeros in the kіltsі, then, otzhe, . From our mirkuvannya weeping like that

Tsya formula є clarified nerіvnostі (5) for vipadku, if kіltsі K has no zeros. Formula (6) is also valid and even if one of the rich terms f(x), g(x) or else is equal to zero. Therefore, the addition of two non-zero rich terms is a non-zero rich term, the following theorem is valid for this:

Theorem 1

We have given an algebraic definition of a polynomial not to avenge the same riddles about functions. Prote, with the skin polynomial over the area of value K, one can associate a natural order with a function that is assigned to K and takes on a value in K.

Come on - a rich member of the coefficients of K. For whatever it is, it's possible

de viraz at the right part is understood as the result of operations at the end of K. Oberzhuvannya at which element is called the value of the polynomial f (x) at the point x0. (The word "speck" is used as an analogy with a drop, if x0 can be represented as a point on the dynamic axis.) In this way, the skin element x0 of the K ring is assigned the element f (x0) of the same ring, and they themselves assign a function on K with the values of K .

It will be shown that the addition of that multiplicity of rich terms is carried out by the most significant operations, which are carried out on functions, if they are added or, obviously, the values of the functions are multiplied at the skin point.

Let's look at two polynomials: , . Let h(x) = f(x) + g(x) – їх sum. Let's say that h(x0)= =f(x0) + g(x0) for skin. Appropriate to the formula (1) = , de , What and it is necessary to bring.

Come on now - additional rich terms f(x) and g(x). Let's say that for the skin. We multiply equivalence , . Koristuyuchis power of operations in kіltsі K (zokrema, commutativeness and associative plurality), we take: , de . The equalization of the taken result with the formula (2) allows the growth of whiskers, which .

In this order, the function, which is represented by the sum (exactly supplementary) of two rich members, is the sum (exactly supplementary) of the functions, which are represented by these rich members.

Seemingly, the similarity between polynomials and their functions is not mutually unambiguous. Prote, since the ring K is inexhaustible, then different polynomials from the ring K[x] must always have different functions.

About leftovers (XTO). Theorem. Come on - pairwise mutually prime the numbers, = , ..., pick up the same 1, = , . Todi solution of the system will look: . Tsya theorem є the basis of the method of orthogonal bases for the hour I will transfer from the system of superfluous classes to the positional number system. Let the foundations of the system of superfluous classes; = = - Set the range of the system. The choice of the system is determined by ...

4. Binary blues. Mathematics as a science creates the world of mutually simple and collapsible objects (speech, phenomena, processes). Abstracting from reality, mathematics looks at unary, binary and other blue. In nutrition, it is necessary to look at the binary blues, their authority especially pays respect to the equivalent equivalence, set on one multiplier. Let's take a look...

X*y. A field is called such an associative commutative ring with unity k, for which there may be some non-zero element animal: . In such a rank, for the appointment of the field, the days of the day are zero. A ring is called an infinity with two operations of the algebra R (+, *), which are: 0. Those elements of an R ring are called reverse, as they can turn around like a multiplication operation, an infinitive R in a given case ...

Quiet, who, having worked at the electrical engineering gallery, began to think about the possibility of creating technology for saving data, which would secure more economical windows of space. One of them was Claude Elwood Shannon, the founder modern theory information. From that time onwards, it was more practical to learn about the algorithms of Huffman and Shannon-Fano embossing. And in 1977. mathematicians Jacob Ziv and Abraham Lempel...

Kіntsі fields can be called up from a bunch of richly divided numbers in the same way, as if fields were called from a bunch of whole numbers. Let me have a ring of richly articulated F[x] over the field F. Just like that, they were prompted for a ring Z, vodnosin ring, you can encourage and vodnosin ring F[x]. Vibelyuchi s F [x] quite rich member p(x), you can name a ring of vіdnosin, vicorist p(x) as a module for the task of arithmetic of the whole circle. We are obsessed by the sight of less than suggestive rich-membered, the shards of the obmezhennya know the inappropriate insignificance of the mind.

Appointment 2.4.1. For a fairly induced rich member p(x) non-zero step over the field F is called impersonality of all polynomials over F, the steps of which are selected from the step of the polynomial p(x), s operations of folding and multiplying rich terms per module p(x). Tse ring is accepted to signify through F(x)/(p(x)).

Additional element r(x) kіltsya F[x] you can visualize the ring element PF[x]/(p(x)) for help r(x)-R P(X). Two elements a(x)і b(x) h F[x], are mapped into one and the same element F[x]/(p(x)), are called equal:

a(x) = b(x)(mod p(x)).

Todi b(x)= Oh)+Q (x) p (x) for deyaky rich member Q(x).

Theorem 2.4.2.Anonymous F1х]/(р(х)) є kіltsem.

Bringing I hope the reader is right.

Vibero in kіltsі richly articulated GF(2), for example, rich term p(x)= x 3+1. Same ring of rich terms per module p(x) one GF(2) [x] / (x 3 + one). It is made up of elements

{0, 1, x, x + 1, x 2, x 2 + 1, x 2 + x, x 2 + x + 1). In this ring, the plural is victorious, for example, in this order:

(x 2 +1) (x 2) = R x 3 + 1 ((x 2 +1) (x 2)) = R x 3 + 1 ((x 3 +1) x + x 2 + x) \u003d x 2 + x,

de vikoristano reduction according to the rule x 4 = x (x 3+ 1) + X.

Theorem 2.4.3.The ring of polynomials modulo the induced rich term p(x) is the same field and only if the rich term p(x) is simple Let's guess that a simple rich term is at once uninducible and inducible. In order to induce the field to prove the irreducibility of p(x), we managed to look at the polynomials less often, and later on, the results would be of a less serious nature) .

Bringing. Give me a rich dick p(x) simple. In order to bring, what is the ring, what is being looked at, making the field, enough to show that the skin of a non-zero element can be a multiplicative return. Come on s (X)-some non-zero element of the ring. Todi deg s (X)< deg p(x). Oskіlki rich member p(x) simple, then gcd = 1. Followed by 2.3.7

NID = 1 =a(x)p(x) + b(x) s(x)

for certain rich members Oh)і b(x). Otzhe,

1 = R p(x)[ 1] = R p(x)= R p(x){ R p(x)

It is easy to bachiti, scho without any other rich members with coefficients in K I make a commutative ring, which is signified k[x] and rank ring of richly articulated over k . Symbol x start to call it “snake”, the terminology of vinyl is clear polynomial functions above R or over C. However, the zagalnomu has a richly articulated polynomial function - the whole speech; for example, over the end field $\mathbb F_p$ from a simple number of elements in rich segments $x$ і $x^p$ set one and the same function, but there are different rich terms (rich terms are considered equal or less than the same, if they have all the coefficients). Otzhe, change x can not enter the overhead field k; about the ring k[x] you can think like this: we add to the multiple elements of the field new element x And just for that, the axioms of China were victorious and x switching from field elements.

The scalars of the elements and the number of rich terms can be multiplied by scalars from the field k, it is actually an associative algebra over the field k. How to look k[x] yak vector space(so forget about the multiplication), there can be an inexhaustible basis with elements 1, x, x 2 etc.

Lay-out at ease k[x]

Factor k[x]
$L \ simeq k [x] / (p).$
An important okremy vipadok - if there is a kіltse, what to avenge k, the field itself; meaningfully yoga K. The simplicity of the factor module by $(p)$ equally strong innocence $p$ . The theorem about the primitive element asserts that, if it is a separable extension, it can be generated by one element, and, therefore, may look at the factor of the ring of rich terms over a smaller field by the rich term, which should not be given. Like a butt, you can bring the field of complex numbers, like generated over R element i, such that i 2 + 1 = 0. Vidpovidno, rich member x 2 + 1 unguided over Rі
$\mathbb(C) \simeq \mathbb(R)[x]/(X^2+1).$
More wildly, for a more complete (navіt non-commutative) circle A what to revenge k that element a kіltsya A, sho komtuє z usima elements k, іsnuє single homomorphism kіletsz k[x] in A, what do you manage x in a:
$\phi:k[x]\A, \quad \phi(x) = a.$
The reason and unity of such a homomorphism is manifested by the additional universal power of the number of richly-membered and explaining the “uniqueness” of the number of richly-membered different designs the theory of ring and commutative algebra.

Modules

A number of richly-membered in the form of a number of changes

Appointment

Rich member vіd n change X 1 ,…, X n with coefficients for field K varies similarly to a polynomial in the form of one change, but the values become foldable. For whatever multi-index α = (α 1 ,…, α n), de leather α i- non-zero integer number, let's
$X^\alpha = \prod_(i=1)^n X_i^(\alpha_i) =X_1^(\alpha_1)\ldots X_n^(\alpha_n), \quad p_\alpha = p_(\alpha_1\ldots\alpha_n)\in\mathbb(K).\$
X α called monomial step $| \ alpha | = \sum_(i=1)^n \alpha_i$ . Rich Member- the last line combination of monoterms with coefficients K: $\sum_\alpha p_\alpha X^\alpha$ .
Rich members of the species n change with the coefficients of the field k(With the most significant operations of folding and multiplying) establish a commutative ring, which is signified k[x 1 ,…, x n]. This ring can be taken away by the bagatorazovannym zastosuvannyam operation "taking a ring of polynomials over this ring". For example, k[x 1 , x 2] isomorphically k[x 1 ][x 2 ], yak i k[x 2 ][x one]. This circle plays a fundamental role in the geometry of algebra. A lot of results in commutative algebra have been achieved to the point of perfecting the ideal ring and modules over it.

Hilbert's null theorem

Dekіlka of fundamental results that stand in mutual connection between the ideals of the country k[x 1 ,…, x n] and algebraic variations k n vіdomі pіd sleeping im'yam Hilbert's zero theorems.

(weak form, closed algebra) Come on k- Algebraic closed field. Then be the maximum ideal m kіltsya k[x 1 ,…, x n] may look

$m = (x_1-a_1, \ldots, x_n-a_n), \quad a = (a_1, \ldots, a_n) \in k^n.$
(weak form, be it a field of coefficients) Come on k- field, K- an algebraically closed field to avenge kі I- Ideal at kіltsi k[x 1 ,…, x n]. Todi I avenge 1 in that and only in that case, if richly articulated I do not make a wild zero in K n .

(strong form) Come on k- field, K- an algebraically closed field to avenge k, I- Ideal at kіltsi k[x 1 ,…, x n] ta V(I) - algebraic subconsciousness, K n pevne I. Come on f- rich term equal to zero at all points V(I). Todі deaky stupіn f live up to the ideal I.

How to win the assignment of the radical to the ideal, tsya theorem f be a radical I. Negative corollary to the form of the theorem - the basis of bioactive similarity between radical ideals K[x 1 ,…, x n] and algebraic variations n-peaceful athenian expanse K n .
Div. also

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Literature

Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings, Berlin, New York: Springer-Verlag , ISBN 978-0-387-95325-0

Lang, Serge(2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag - ISBN 978-0-387-95385-4 , MR1878556

Osborne, M. Scott (2000), Basic homological algebra, Vol. 196, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98934-1

Lesson that characterizes the ring of rich-membered
- Where to lay your head? - after sleeping Mykola, pod'zhdzhayuchi krokiv a hundred to the thought that he suspected. Ale did not catch the thought of seeing, like a rusak, watching the frost until tomorrow's morning, not hanging and gathering. Zgraya hounds on bows, with a roar, rushed up the mountain after a hare; from the sides of the horti, which were not on the ground, they rushed to the hounds and to the hare. All the thinkers, who were collapsing enough, the vizhlyatniks shouting: stop! beating dogs, greyhounds shouting: atu! directing the dogs - they galloped across the field. Calm Ilagin, Mykola, Natasha and the uncle flew, they themselves did not know how and where, bachachi only dogs and a hare, and they were afraid only to waste their money on a mile of sight. Hare caught in matter and chewing. Having huddled together, galloping wildly, and povіv vuha, listening to the cry and stupidity, which raptom pierced from the sides of the mouth. Having cut ten times without a hitch, letting the dogs reach you, and nareshti, vibrating straight ahead and realizing the insecurity, pecking your ears and rushing your mustaches. Vіn lying on the stubble, but in front of the boulders were green, for which it was swampy. Two dogs of a suspicious mind, which were the closest, first marveled and slaughtered for a hare; And yet they didn’t go far to the new one, as because of them Yerza, the Ilaginskaya red-footed dog, flew up, approached the dog in front of me, with a terrible whiplash, aimed at the hare’s tail and thinking that she had scrambled yogo, slumped around the sack. Hare wiggling your back and pushing even harder. Because of Yerzi, she was wide-assed, the black-haired Milka and Shvidko began to sleep until the hare.
- Nicely! matinko! - felt the triumphant cry of Mikoli. It was supposed to hit Milk at once and drink the hare, but then she overtook and rushed off. Rusak vіdsіv. The beauty Yerza has again attacked and hung over the very tail of the hare, no matter how reconciled, don’t have mercy now, go to the back of the quilt.
- Yerzanko! sister! - I felt the weeping, not my own voice of Ilagin. Yerza did not feel yoga blessings. At that very moment, it was like a check was needed, that a hare would go out, vіn vyhnuv and vykotiv on the border between greenery and stubble. I know Yerza and Milka, like a dishlava couple, they woke up and began to fall asleep until a hare; at the border it was easier for the hare, the dogs did not approach so quickly to the new one.
- Lai! Laika! March on the right! - shouting in a new voice, and Scolding, red, humpbacked dog, uncle, wriggling and twisting his back, staring at the first two dogs, hanging out behind them, pressing from a terrible self-confidence over the hare itself, beating yogo from the cordon on the greens , another time, having pressed angrily on the wild greens, stomping on the knee, and only a boulder can be seen, like a vein of strimgols, wandering back into the ford, swaying like a hare. Zirka dogs perfected yoga. Through the khvilina, everyone stood white of the dogs that stood up. One happy uncle was crying and resigned. Shaking a hare, shedding blood, looking around anxiously, eyes wide, not knowing the camp of her hands and feet, and saying, she herself did not know what it was.
“Otse march on the right ... the axis of the dog ... the axis of pulling the shoulders, and the thousandths and the ruble - the march is clean on the right!” saying wine, gasping and looking around angrily, no barking at anyone, no all were foes of it, all of it was imagining it, and only now you have come to tell the truth in the distance. "Axis to you and a thousand - a clean march on the right!"
- Lay, go to bed! - saying wine, throwing a paw with sticky earth; - Deserving - a clean march on the right!
- Vaughn waved, gave three days alone, - showing Mykola, also not listening to anyone, and not worrying about those who listen to yoga, chi.
- That tse scho is across! - saying Ilaginsky stirrups.
“So, as if you’ve slumped, so every doorkeeper sleeps like this,” Ilagin, chervony, speaking at once, heavily translating his breath from the haircut and praise. At the same hour, Natasha, without taking her breath away, squealed radiantly and hoarsely, so piercingly that it rang at the vuhah. Vaughn hung with her temple all those who hung their other thoughts with their one-time rose. And this heather is so wondrous, that she herself would be small enough to litter this wild heather and all the fault would be to rise to you, it was like a boulder at this hour.
Dyadechko himself echoed the hare, quietly and chewingly throwing yoga over the back of the horse, nіbi dorikayuchi to all of them, and with such a look that you don’t want to speak wine and kim, sіv on your kaurago and go get out. Everything, around the new, summative image, rose up and only a long time later could come to a colossal indentation of canopy. For a long time the stench looked at the red-faced Rugai, who, from a straggling brook, with a humpbacked back, a clinking bay, with a calm look, overcame the uncle behind the legs of the horse.
“Why am I so myself, like everything else, if you don’t bump to the right to squash. Well, get over here! it was given to Mikoli, saying that he looked like a dog.
If, long ago, the uncle went to Mikoli and spoke to him, Mikola would be pleased with him, that, uncle, after all, that it was, that I would deign to speak with him.
If in the evening Ilagin said goodbye to Mykola, Mykola leaned on such a far-off station to the booth that, having adopted the proposition of the uncle, he would deprive him of sleeping with him (at the uncle), at the yogi’s village of Mikhailivtsi.
- I yakbi came before me - the march is clear on the right! - having said uncle, it would be better; Bachite, the weather is wet, having shown the uncle, they would have rested, they would have taken the countess in a droshky. - The proposition of the uncle was accepted, for the droshky they sent a messenger to Vidradny; and Mikola, Natalka and Petya went to their uncle.
A man to five, great and small, yard cholov_kiv vibіg on the front ganok of the master. Dozens of wives, old, great and small, hung from the back of the ganka marveling at the wise men, who were crying. The presence of Natasha, the women, the tops, brought the clamor of the yard uncles to the quiet between, richly, not quarreling at the presence, went up to her, looked into her eyes and in front of her robbed her of their respect, like a miracle, like not showing up, like not a human being, and you can’t think a little about what to talk about him.
- Arinko, look, sit on the barrel! To sit on her own, but podіl bovtaєtsya... Bach rіzhok!
- Fathers of light, then a knife ...
- Bach Tatar!
- How did you not spread? - said the naismilivisha, straight ahead as far as Natalka.
The uncle's horse beat the gank of his wooden budinochka overgrown with a garden, and looking around his household, shouting at the command, so that they came in and that everything necessary for the reception of the guests was broken.
Everything broke up. Dyadechko, having taken Natasha from his horse and by the hand, took a hank along the cunning boardwalks. At the booth, not plastered, with broken walls from the logs, it was not even clean - it was not visible, so that the meta people lived, I thought that there were no flames, but there was not a lot of zanedbanosity.
The blue trees smelled of fresh apples, and hung owls and fox skins. Through the front uncle, we see our guests at a small hall with a folding table and red stilts, then at the vital room with a round birch table and a sofa, then at the office with a torn sofa, we put on a kilim and portraits of Suvorov, father and mother of the ruler and yoga himself in the military uniform . . There was a strong smell of tyutyun and dogs near the office. At the office, the uncle asked the guests to sit and roam like at home, and he himself wiyshov. Barking with your back, that you haven’t cleaned yourself, you’ve gone to the office and lie down on the sofa, cleaning yourself with your tongue and teeth. From the office there is a corridor, at which one could see screens with torn flanks. Behind the screen, I felt a woman’s laugh and whisper. Natasha, Mikola, and Petya sprang up and sat on the sofa. Petya leaned on his arm and immediately fell asleep; Natasha and Mikola were sitting together. Their appearances were burning, the stench was more hungry and more cheerful. The stinks marveled one on one (since polyuvannya, at the kіmnati, Mykola already did not care for the need to show his human dignity in front of his sister); Natasha blinked at her brothers, and the insults abated for a short time, and they burst out crying, unable to come up with excuses for their laughter.
Trochs of the uncle's eyes were taken away by the cossacks, blue pantaloons and little boots. І Natasha thought that this was the very suit, for which the uncle in Otradnoye drank with wonder and gluzuvannyam - it was a right suit, which was not worthy of anything for surduki and frocks. Uncle buv tezh cheerful; not only did he not form the laughter of his brother and sister (it could not occur to you that they could laugh at his life), but he himself came to their causeless laughter.
- The axis is so young the countess - the march is clean on the right - the other one is not so bachiv! - Having said wine, giving one pipe with a long chibouk to Rostov, and another short, cutting chibouk laying with a solemn gesture between three fingers.
- Day vіd'їzdila, hoch cholovіkovі at the right time and like no matter what happened!
Suddenly, after a while, the uncle opened the door, by the sound of her, the girl was clearly barefoot, and in the door with a large furnished dance in her hands, a tovsta, a rum'yana, appeared, garna zhіnka rokіv 40, with podvіyny pіdborіddyam, and povni ruddy lips. Vaughn, with a drawing-room representation and a splendid look in her eyes and a skin-like Russian, glanced over at the guests and with a sly grin bowed down to them. Irrespective of the comrade, the lower voice, which zmushuvala її put forward the chest and live and back trim the head, the woman (the uncle's housekeeper) stepped superbly lightly. Vaughn walked up to the table, set the table, and with her big hands, took it with her plump hands and placed the dances on the table, appetizers and chastuvannya. After finishing it, she came out and with a smile on her face became the white of the door. - “Axis out and me! Now you understand, uncle? told Rostov її appearance. How not to understand: not only Rostov, but Natasha, the uncle understood the meaning of frowning brows, and a happy, self-satisfied smile, like a troch wrinkled yoga ruin at that hour, as Anissa Fedorivna entered. There were herbs, liqueurs, mushrooms, shortbreads of black boroshna on yura, stylnikovy honey, honey of jams and effervescent, apples, peas and peas in copper. Then they brought Anisya Fyodorivna and boiled on copper and on zukri, and a tare, and a cock, thoroughly greased.