Linear Algebra: Chapter 1 polynomial 1

§ 1 Field

The addition, subtraction, multiplication, division, and other operations of numbers are generally called the algebra of numbers. the problem of algebra mainly involves the algebraic nature of numbers. Most of these properties are common to all rational numbers, real numbers, and plural numbers.

Definition 1It is a set composed of multiple numbers, including 0 and 1. if the sum, difference, product, and Quotient of any two numbers are still the same number, it is called a number field.

Obviously, a set composed of all rational numbers, a set composed of all real numbers, and a set composed of all plural numbers are all Number Fields.Q, R, CTo represent. A set of All integers is not a number field.

If the result of an operation on any two numbers in a data set is still in the middle, the number set is closed for this operation. therefore, the number field definition can also be said that if a number set, including 0, 1, is closed for addition, subtraction, multiplication, and Division (Division not zero), it is called a number field.

Example 1All Forms

Number (any rational number), which forms a number field. It is usually used to represent this number field.

Example 2All Forms Available

Is an integer.

Example 3All number sets composed of odd numbers are closed for multiplication, but not for addition or subtraction.

Nature: All vertices contain rational vertices as part of them.

 

 

 

 

§ 2 unary Polynomials

1. Polynomial

Definition 2It is a non-negative integer. It is a formal expression.

, (1)

All of them belong to the number field, which is called the one-dimensional polynomial of the coefficient in the number field or the one-dimensional Polynomial in the number field.

In polynomial (1), it is called a sub-term, called a coefficient of a sub-item. Later, it is expressed as a polynomial by or.

Note:: The polynomial defined here is a form expression of symbol or text.

Definition 3If, except for the zero coefficient, the coefficients of the same term are all equal in polynomial and medium, it is equivalent to it, and is recorded.

A Polynomial with zero coefficients is called a zero polynomial and is counted as 0.

In (1), if it is called the first item of Polynomial (1), it is called the first item coefficient and the number of times of Polynomial (1. zero polynomials are the only polynomials that do not define the number of times. the number of polynomials is recorded.

Ii. polynomial operation

Set

It is two polynomials in the number field.

In representing polynomial and time, for example, for convenience, in order, then sum

The product of sum is

The coefficient of the sub-item is

Therefore, the table can be

.

Obviously, after the two polynomials in the number field are calculated by addition, subtraction, and multiplication, the result is still a polynomial in the number field.

It is not difficult to see the addition and subtraction of polynomials.

.

For polynomial multiplication, we can prove that, if, then, and

From the above proof, we can see that the first coefficient of the polynomial product is equal to the product of the first coefficient of the factor.

Obviously, the above results can be generalized to multiple polynomials.

Polynomial operations meet the following rules:

1. addition exchange law :.

2. Addition law:

3. Multiplication exchange law :.

4. Multiplication law:

5. Multiplication:

6. Multiplication elimination law: If and, then.

Definition 4All coefficients are the whole of the unary polynomials in the number field. They are called The unary polynomial rings in the number field.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

§ 3 concept of Division

In a polynomial ring, addition, subtraction, and multiplication can be performed. However, inverse multiplication-division-is not common. the Division is a special relationship between two polynomials.

I. Concept of Division

Division with RemainderFor any two polynomials and, among them, a polynomial must exist

(1)

Or, and this is the only decision.

The operators obtained in the Division with remainder are usually called operators of division, and they are called operators of division.

Definition 5The polynomials in the number field are called the Division. If there is a polynomial in the number field that makes the equation

True. Use "" to indicate division, and use "" to indicate division failure.

At that time, it was called the formula, called the double formula.

At that time, the Division with remainder provided a discriminant condition for the Division.

Theorem 1For any two polynomials in the number field, the necessary and sufficient condition is that the remainder formula is zero.

The remainder Division must not be zero. But the center can be zero. At this time.

At that time, for example, Division vendors sometimes used

.

Ii. Division nature

1. Any polynomial must divide itself.

2. Any polynomial can divide zero polynomial 0.

3. Zero polynomial, that is, a non-zero constant, can divide any polynomial.

4. If yes, the non-zero constant is used.

5. If so, (the transfer of the entire Division ).

6. If, then

,

It is any Polynomial in the number field.

Generally, it is called a combination.

From the above properties, we can see that there is the same formula as any of its non-zero constant times, because it can often be used instead in the discussion of Polynomial divisible.

Finally, the Division relationship between two polynomials does not change because of the expansion of the number field. that is, if it is two polynomials in the middle, it is a large number field. of course, it can also be seen as a polynomial in. it can be seen from the Division with remainder that, whether viewed as a polynomial in or in the middle, the formula used to remove the formula is the same as the formula used to remove the formula. therefore, if Division is not allowed in, division is not allowed in.

Example 1Verify if, then

Example 2Please make.

Example 3If yes, then.

§ 4. Maximum formula of Polynomial

1. Maximum formula of Polynomial

If the polynomials are both causal and causal, it is called a common formula of the sum.

Definition 6Let's assume that it is a common formula of two polynomials in which the polynomial is called, if it meets the following two conditions:

1) It is a public factor;

2), the Internet is all the Internet.

For example, for any polynomial, It is a maximum formula of 0. In particular, according to the definition, the maximum formula of two zero polynomials is 0.

TheoremIf there is an equation

(1)

Yes, so, And, have the same Internet.

Theorem 2For any two polynomials, there is a largest formula in, and can be expressed as a combination of, that is, a polynomial

. (2)

It is not difficult to see from the definition of the largest formula that, if there are two largest ones, there must be a sum, that is. that is to say, the maximum formula of two polynomials is uniquely determined in the sense that a non-zero constant multiple can be different. the maximum formula of two completely zero polynomials is always a non-zero polynomial. in this case, we agree to use

(,)

To indicate the largest formula of 1 for the first coefficient.

The method used in theorem proof to obtain the maximum formula is usually called division algorithm ).

ExampleSet

Seek (,) and make

.

Note:The inverse of Theorem 2 is invalid. For example

,

Then

.

But it is obviously not the biggest Internet factor.

However, when the formula (2) is established, but is a public factor, it must be the largest public factor.

Ii. Polynomials

Definition 7The two polynomials in the table are called the mutual Prime. If

Obviously, there is no other formula except the zero Polynomial for the two polynomials, and vice versa.

Theorem 3
There are two polynomials in, and the necessary and sufficient condition for the reciprocal is that there is a medium polynomial.

.

Theorem 4If, and, then

.

Inference 1If, and, then

.

Inference 2If, then

Promotion: For any number of polynomials, a maximum formula is called. If it has the following properties:

1 );

2) if, then.

We still use symbols to represent the largest formula with the first coefficient of 1. It is not hard to prove that the largest formula exists, and when all is not zero,

Is the maximum Internet factor, that is

=

Similarly, the above relationship can prove that there is a polynomial

If so, it is called the reciprocal element. There are also conclusions similar to Theorem 3.

Note:1) when a polynomial divisible the product of two polynomials, if there is no reciprocal condition, this polynomial generally cannot be one of the operators of the divisible product. For example, but, and.

2) if there is no mutual element in inference 1, it is not true. For example ,,

,.

Note:When there are polynomials, they are not necessarily two-to-one. For example, Polynomials

They are mutually vegetarian,.

This is a number field. It is the greatest formula of the polynomial with the first coefficient of 1 in the column, but the largest formula with the first coefficient of 1 in the column.

That is to say, when the transition from a number field to a number field, it does not change in nature with the maximum Internet factor.

The properties of reciprocal polynomials can be extended to multiple polynomials:

1) if the polynomials and

, Then.

2) If the polynomials are all divisible, and the two are mutually exclusive, then.

(3) If the polynomials are all mutually exclusive

.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

§ 5 factorization theorem

1. Non-approximately Polynomials

.

Definition 8The polynomial of the number of times in the number field is called the ircomprescible polynomical in the field. If it cannot be the product of the polynomial of the number of times in the number field.

According to the definition, a polynomial is always a non-approximate polynomial.

Whether a polynomial can be reduced depends on the system number field.

Obviously, there are only two kinds of causal formulas of non-zero constants and its own non-zero constants. in turn, the polynomial of the number of times with this property must be non-correlated. from this we can see that there may be only two relationships between non-approx polynomials and any polynomial, or.

Theorem 5If it is a non-approximate polynomial, then for any two polynomials, it must be introduced or.

Promotion: If the product of some polynomials cannot be divisible by the non-approximate polynomials, one of these polynomials must be divisible.

Ii. factorization theorem

Factorization and Uniqueness TheoremThe number of polynomials in the number field can be uniquely decomposed into the product of some non-approx polynomials in the number field.

,

Then there will be, and there will be an appropriate order after

.

There are some non-zero constants.

It should be pointed out that although the factorization theorem has its basic importance in theory, it does not provide a specific method for decomposing polynomials. in fact, in general cases, the commonly feasible method for decomposing polynomials does not exist.

In the factorization formula of polynomials, we can propose the first coefficient of each non-repudiation formula to make them a polynomial with the first coefficient of 1, and then merge the same non-repudiation formula. the decomposition form becomes

,

Where is the first coefficient, which is a positive integer rather than an approximate polynomial with different first coefficients of 1.

If there is a standard decomposition of two polynomials, we can directly write out the maximum formula of two polynomials. the greatest common formula of a polynomial is the product of an unmodifiable polynomial power that occurs simultaneously in the standard factorization formula, the exponent of the power brought is equal to a smaller one in the power brought in.

From the above discussion, we can see that the Division with remainder is the basis of the theory of Factorization of a polynomial.

If there is no common non-approximately Polynomial in the standard factorization formula, it is the same as the reciprocal Prime.

Note:: The method for finding the maximum formula above cannot replace the moving phase division, because under normal circumstances, there is no actual method for decomposing polynomials as the product of non-approx polynomials, it is usually difficult to determine whether a polynomial on the number field is allowed.

For example, in the rational number field, the product of the polynomial is decomposed into non-approximate polynomials.