Expand the polynomial over the field of real numbers. The numerous joints are pointed and not pointed. Rich terms over the field of rational numbers
A term over a ring of integers is called primitive, since the largest negligible part of its coefficients is equal to 1. A rich term with rational coefficients is uniformly represented by the addition of a positive rational number, which is called zmіstom rich member and primitive rich member. The addition of primitive rich members is a primitive rich member. This fact implies that a polynomial with integer coefficients is imposed over the field of rational numbers, and it is imposed over the ring of integer numbers. In this way, the task of expanding a polynomial into multipliers so that it does not work over the field of rational numbers is reduced to a similar problem over the ring of integers.
Let - a rich term with whole coefficients and in place of 1, and - its rational root. Let's imagine the root of a polynomial as a fraction. rich member f(x) appears to be the creation of primitive polynomials. Otje,
A. number manager and business manager,
B. znamennik – dilnik
C. for any whole k significance f(k) – an integer number that can be divided without excess by ( bk-a).
Listings of power allow you to inform the task of identifying the rational root of the rich member until the end search. A similar approach can be used for the expanded polynomial f on multipliers, so as not to navigate over the field of rational numbers using the Kronecker method. Is it a rich member? f(x) step n obviously, then one of the multiples may have a step no higher n/2. This multiplier is significant through g(x). Since all the coefficients of polynomials are integers, then for any whole a significance f(a) to be divided without surplus into g(a). Vibemo m= 1+n/2 different whole numbers a i, i=1,…,m. For numbers g(a i) there is a final number of possibilities (the number of members of any non-zero number is essential), also, there is a final number of rich members that can be partners f(x). Having carried out a further search, we will either show the innocence of the rich member, or we will separate them into two rich members. Before the skin multiplier, the scheme is clearly indicated until all the multipliers become rich members, so they are not induced.
The irreducibility of real polynomials over the field of rational numbers can be established using a simple Eisenstein criterion.
Let's go f(x) a polynomial over the ring of integers. What is a simple number p, what
I. All coefficients of the polynomial f(x), in addition to the coefficient at the senior level, are divided into p
II. The coefficient for the senior level is not divided into p
III. A valid member does not belong to
Todi is a rich member f(x) is non-guided over the field of rational numbers.
It should be noted that Eisenstein’s criterion provides sufficient evidence for the non-reduction of rich members, but is not necessary. Thus, the rich term is not applicable over the field of rational numbers, but does not satisfy the Eisenstein criterion.
The rich term, according to Eisenstein’s criterion, is not inducible. Well, over the field of rational numbers there are non-reducible terms of a large degree n, de n be the natural number greater than 1.
The field is called closed algebra, since any polynomial over this field that is not equal to a constant may have at least one root. From Bezout's theorem it immediately follows that over such a field, any non-constant rich term can be expanded into three linear multipliers. In whose sense the algebraically closed fields are simpler, less algebraically closed. We know that over the field of real numbers there is no square trinomial with a root, so the field ℝ itself is not a closed algebra. It turns out that it doesn’t take long until the algebra is closed. In other words: those who lived would have secretly spoken about the rivalry in private, but we would immediately run into the gap of the polynomial rivalries.
FUNDAMENTAL THEOREM OF ALGEBRI. If any polynomial over the field ℂ is not equal to a constant, there may be one complex root.
EVIDENCE. Any polynomial, not equal to a constant, over the field of complex numbers can be expanded to the addition of linear multipliers:
Here is the senior coefficient of the polynomial, - all the different complex roots of the polynomial, - their multiplicities. It is the fault of jealousy
The proof of the corollary is a clumsy induction behind the step of the rich term.
Over other fields, the situation is not so good in the sense of the layout of the rich members. We say that the rich term is not identifiable, since it is, first of all, not a constant, but, in other words, it cannot be decomposed among the rich members of lower levels. It is understood that any linear polynomial (over any field) should not be induced. The result can be reformulated as follows: the unreduced rich terms over the field of complex numbers with a single leading coefficient (aka: unitary) are drawn from the rich terms of the form ().
The foldability of a square trinomial is equal to the presence of at least one root. Converting the equation to the view, it can be concluded that the root of a square trinomial is the same if the discriminant is the square of any element of the field K (here it is assumed that 2≠ 0 for the field K). Stars can be removed
PROPOSITION, REQUEST, REQUEST. Squarey trichs above the field K, yaku 2 ≠ 0, confessively to the tilki tilki, if it is not a lot of Korinnya in half a k. Tselno is not a cerebral of the Hyodnoye Elentent field K. Zokremama, above the field of the quadratic trichlen , like and only, like.
Now, above the field of real numbers, there are two types of terms that are not visible: - linear and quadratic and negative discriminant. It turns out that these two types of attacks involve the absence of many joints that are not directed over ℝ.
THEOREM. Any polynomial over the field of real numbers can be expanded into linear multipliers and quadratic multipliers with negative discriminants:
Here are all the different active roots of the rich term, their multiplicities, all discriminants are less than zero, and the square trinomials are all different.
We'll let Lema know right away
LEMA. If anyone has one, then the resulting number is also the root of the polynomial.
Finished. Let go and the complex root of the rich member. Todi
where we vikoristed the authorities from the receipt. Otje, . Tim himself is the root of the rich member. □
Proof of the theorem. It is enough to prove that any rich term that cannot be reduced over the field of real numbers is either linear or quadratic with a negative discriminant. Let's not bring a rich member with a single senior coefficient. Sometimes it’s impossible to do something active. Let's say, okay. It is significant through any complex root of this rich term, which underlies the main theorem of the algebra of complex numbers. The fragments are irreducible, then (the marvelous Bezout theorem). So, according to Lemi, it will be another root of the rich member, edited out.
A rich member has an effective coefficient. In addition, it is reasonable to compare with Bezout’s theorem. Since there is a single senior coefficient inexplicably, then jealousy is eliminated. The discriminant of this polynomial is negative, as it is also in bi speech roots.
APPLY. A. Let's decompose the polynomial into multipliers so as not to overwhelm. Among the members of the constant term 6 we find the root of the rich term. Let's convert it so that 1 and 2 are radicals. Tim himself is a rich member. Having shared, we know
The remainder of the expansion over the field, since the discriminant of the square trinomial is negative and, therefore, cannot be expanded further over the field of real numbers. The expansion of the same rich term over the field of complex numbers can be eliminated once we know the complex root of the square trinomial. The stench is the essence. Todi
Unfolding this rich member over
B. Decomposition over the fields of active and complex numbers. Since this rich term does not have active roots, it can be decomposed into two square trinomials with negative discriminants
Since when replacing with a rich term it does not change, then with such a replacement the quadratic trinomial must be transferred back and forth. Zvidsi. Equivalent coefficients when under control, sedation, . From the relationship (by substitution it is possible, and residually, .
Lay out over the field of active numbers.
In order to decompose this polynomial over complex numbers, it is necessary to compare. It dawned on me what will happen to the roots. All different roots are removed from us. Otje,
Solving problems with complex numbers. Easy to calculate
And we find another solution to the problem of expanding the rich term over the field of real numbers.
The end of the robot -
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N.I.Dubrovin
Spaske Gorodishche 2012 Zmist Intro. 4 List of meanings of terms. 5 1 Little things about BASIC. 6 2 Naive theory of multiplicities. 9
Little things about BASIC
Mathematics can deal with such objects as numbers of different nature (natural, purposeful, rational, efficient, complex), many terms of one and many variables, matrix
Naive theory of multiplicities
A mathematical text is composed of values and solids. Actions of insistence on importance and importance and commitment to other affirmations are called one of the following terms:
Cartesian creations
A pair, or simply a pair of elements, is ordered, which is one of the fundamental constructions in mathematics. You can imagine her as a friend of two places - the first and the other. It’s not very common in mathematics
Natural numbers
Numbers (1,2,3, ...) that can be subtracted with one addition operation are called natural numbers and are designated ℕ. An axiomatic description of natural numbers can be like this (div.
Recursion
From axioms N1-N3 to those familiar to everyone from the cob school, the operation of addition and multiplication of natural numbers, the equalization of natural numbers among themselves and power in the form of "changes in the place of dodankov sums are not
Order on impersonal natural numbers The completeness of natural numbers Completeness of whole numbers Euclid's algorithm Matrix interpretation of the Euclidean algorithm Elements of logic Vyslovluvalny forms Matrix algebra Officials Linear transforming planes Complex numbers Construction of the field of complex numbers Finding complex numbers Trigonometric form for writing complex numbers Complex exponent Unraveling of square plains EQUIVALENCE THEOREM Non-reducible term- a rich member, not decomposed into non-trivial rich members. Numerous terms are not given with elements that are not given, rings of polynomials. A non-reducible multi-term over a field is a multi-term The rich term f over field F is called unprejudiced (sorry), since it has a positive step and does not have non-trivial partners (either any partner or association with him or with one) Proposition 1 Let's go R- Non-drive i A- Be a polynomial of the ring F[x]. Todi chi R divide A, or Rі A- Forgive each other. Proposition 2 Let's go f∈ F[x], i stage f = 1, therefore, f is a rich term that cannot be induced. For example: 1. Take the polynomial x+1 over the field Q. This step is higher than 1, so don’t go there. 2. x2 +1 – non-guided, because the root doesn't matter SLU. Version of the system. Sleepy, crazy, songs and unimportant systems. Equivalent systems A system of linear alignments over the field F with variable x1, xn is called a system in the form A 11
X 1
+ … + a 1n x n= b 1
……………………….. a m1 x 1
+ … + a mn x n= b m de a ik,b i∈ F, m is the number of equals, and n is the number of unknowns. Briefly, the qiu system can be written as follows: ai1x1 + … + a in x n= b i (i = 1,…m.) This SLU is a washroom with n free changeable x 1, .... Xn. The SLNs are divided into nonsense (cannot make a decision) and spilni (songs and insignificance). The sleeping system is called the singing system, because it has a single solution; If she has two different decisions, then she is called undecided. For example: above the Q field x + y = 2 - absurd system x – y = 0 - full song (x, y = ½) 2х + 2у = 2 – bedroom unimportant Two systems L.U. They are equivalent, since the solutions of these systems are avoided, so that the solutions of one system are simultaneously the solutions of others. The system equivalent to the data can be derived: 1. replacing one of the equals of this equals, multiplying the number by substituting zero. 2. replacing one of the levels with the sum of the other levels of the system. The solution of SLE is performed using the Gaussian method. 45* Elementary re-creation of systems of linear ranks (slu). Gauss method. Def.By elementary re-creations of S.L.U n-sya such re-creations: 1. Multiplying one of the system levels of the system by a non-zero element of the field. 2. Additions to one from the level of the system of another level, multiplied by the field element. 3. Adding to or switching off from the system a non-zero level 0*х1+0*х2+…+0*хn=0 4.
Changing places of equals SuggestionLet the system (**) be removed from the system (*) for the additional end number. Element-their transformation. Todi system (**) ~ system (*). (No document) Deputy When recording a system of linear levels, matrix recording is used. a11 a12 … a1n b1 a21 a22 ... a2n b2 ………………….... …
Am1 am2 ... amn вn Butt: 1) 2x1 - x3 = 120-11 x1 - x2 - x3 = 0 1 -1 -1 0 3x1 + 2x2 + 4x3 = 2 3 2 4 2 2) 1 0 1 x1 = 1 0 1 2 x2 = 2 3) 1 0 1 2 x1+x3=2 x1=2-x3 0 1 -1 3 x2-x3 = 3 x2 = 3 + x3 Gauss method Suggestion Let the system go (*) (a) if all free members agree 0 all in = 0 number of decisions = F n (b) k vk = 0 0x1 + 0x2 + ... + 0xn = vk = 0 (no solution) 2.
not all aij=0 (a) in the system there is a level of the form 0x1 + 0x2 + ... + 0xn = vk = 0 0 (b) there are no such ranks b1. Including non-zero level. We know the smallest index i1, such that not all coefficients exist at xij = 0. 0……0……….. …. Another set of zeros is i1. 0……0…..*=0….. …. 0……0 ...……… … 1. by rearranging the levels it is possible to achieve a1i1 = 0 0….. 0… a1i1 = 0….….(1). :=(attached) (1) 1/ a1i1 (2). :=(2)-(1)* а2i1 A2i1........... .... 0…. 0…1…. …. 0…. 0..1….. ….. ( often 0…. 0… а2i1… 0…..0..0… …. Matrix) 0 ........... 0 .... ami1.. ... ……………… …. …………………… …. 0 ….0 ..ami1 ... 0……0…………0 …. Through the final number of cuts we can remove the system to match the form 0х1+0х2+…+0хn=вк=0 0or 0……0 1………….. L1 “direct Gauss move” 0....0 1...0..0 .....0........0.... .. "The Gateway 0......0 0......1..... L2 0....0 0.....1.........0.... .....0.... ..Gauss” 0 .......00.......0....1 L2 0....0 0......0........1... ......0.... .. .............................. .... ............................................ .. 0........0 0 ............0..1 Lk 0....0 0.......0....... ..0....0.......1 .. Zmіnnі xi1, ...... xik are called heady, reshta free. k=n => c-a song k 2 0 -1 1 8 (-3) 1 -1 -1 0 *(-2) 1 -1 -1 0 1 -1 -1 0 ~ 2 0 -1 1 ~ 0 2 1 1 3 2 4 2 3 2 4 2 0 5 7 2 Over the field of real numbers, any irreducible polynomial of one variable is of stage 1 or 2, and the polynomial of the 2nd stage is not directable over the field R either or only if there is a negative discriminant, for example, the polynomial is not directable o over the field of real numbers, fragments of its discriminant negative. The Emisenstein criterion, a sign of the innocence of a rich member, is named after the German mathematician Ferdinand Eisenstein. Regardless of the (traditional) name, it is itself a sign that it is sufficiently intelligent - but not at all necessary, as one might assume, coming from the mathematical substitution of the word “criterion” Theorem (Eisenstein criterion).
Nehai is a rich term over the factorial ring R ( n>0), and any irreducible element p come to this conclusion: Does not share with p, Share on p for whatever reason i view 0
before n- 1, Chi is not divided into. Todi rich member indirectly over F private ring field R. Investigation. Over any field of algebraic numbers there emerges the non-reducibility of a large term of any predetermined level; for example, rich member, de n>1 ta pЇ deyake is a simple number. Let's look at the application of this criterion, if R is the ring of integers, and F is the field of rational numbers. Apply it: The rich term should not be pointed over Q. Do not aim the penis under the stake. True, if it is induced, then the rich term is induced, and the remainder of all coefficients, except the first, are binomial, so that they can be divided by p, and the remaining coefficient is amen p And before that, do not divide yourself into those according to Eisenstein’s criterion, and do not be overly biased. The next five rich members demonstrate the actions of the elementary power of rich members, which is not induced: Above the ring Z of integers, the first two terms are pointed, the remaining two are not pointed. (The third word is a polynomial over whole numbers). Above the field Q of rational numbers, the first three rich terms are guided, the other two are non-guided. Above the field R of real numbers, the first few terms are shown or not shown. The field of real numbers has non-directional linear terms and quadratic terms without real roots. For example, the expansion of a polynomial in the field of real numbers looks like. The offending multipliers in this layout are rich in terms, so don’t be induced. Above the field C of complex numbers, five rich terms are shown. In fact, the same type of constant polynomial over C can be factorized in the form: de n- step of the penis, a- Senior coefficient, - Root of the polynomial. Therefore, the same irreducible terms are rich in linear polynomials (the main theorem of algebra).
For rich people there is a linear order. Let's say that n
The same operation is always possible in the sphere of natural numbers. This gives us the right to introduce a divisibility relation: let's say that the number n divides the number m, since m=nk for any kind of k∈
Significantly through - the ring of integers. The term “ring” means that we are located to the right of the anonym R, in which there are two operations – addition and multiplication, which are ordered by the same rights.
Given a pair of integers (m,n). It is important that n is a surplus with the number 1. The first step to the Euclidean algorithm is to divide m by n with a surplus, and then divide the surplus by a surplus, so that we can make a new addition, while still removing the new
Let's give a matrix interpretation to the Euclidean algorithm (coming paragraph about div matrices). Let's rewrite the sequence of subdivisions from the excess matrix view: Substances in the skin
Mathematicians work right with objects such as, for example, numbers, functions, matrices, lines on a plane, etc., and also work right with formulas. Vyslovlyuvannya is the act of confirmation
Chi viraz bude vislovlyuvannyam? No, this entry is in a single-change form. If instead of changing the substitute for acceptable values, then the differences are clearly defined as
Matrix algebra over the ring R (R is the ring of integer numbers, the field of rational numbers, the field of real numbers) is the most widely studied algebraic system with an immutable operation.
The value of the square matrix A is the numeric characteristic that is indicated. Let's look at the matrix of small dimensions 1,2,3: VALUE. Pu
It is clear that, be it the transformation of the plane, it saves the distance, either parallel transfer to the vector, or rotation around the point Pro on the cut α, or the symmetry is clearly straight
This section has only one field - the field of complex numbers ℂ. From the geometric point of view, vono is a plane, and from the algebraic point of view, in the center
We actually already knew the field of complex numbers in the previous paragraph. Looking at Vinyatkov, the importance of the field of complex numbers is induced by his non-median construction. Let's take a look at the space
The field of complex numbers gives us a new power - the manifestation of a non-identical, uninterrupted automorphism (isomorphism of wines). A complex number is called the one obtained before, and the display
A complex number is conceivable as a vector. Dovzhin's vector, that is. the quantity is called the modulus of a complex number and is designated. The value is called the norm of the number, sometimes it is more difficult to define it
Rule (2) of paragraph gives us the right to value the exponent of a purely explicit number: Indeed, in this way the singing function has such power: &
A linear polynomial always has a root. The square trinomial always has a root over the field of real numbers. Let's find a quadratic trinomial over the field of complex numbers (). Convoy
Let “ ” mean equivalence on the multiplicity M. For an element it is significant through the equivalence class. Todi impersonal M is divided into uniform classes of equivalence; skin element z M with
changes above the field and a simple element of the ring 
, then, unrepresented in the appearance of the work, where i are rich in terms with coefficients, substituted for constants.