Section 5: Divisibility in F[x], greatest common divisor, and the Euclidean Algorithm

HW p. 14 # 1-6 at the end of the notes

In this section, we examine the divisibility of polynomials more closely and examine what is meant by the greatest common divisor of two polynomials.

Divisibility of Two Polynomials Over a Field

Recall the division algorithm for the polynomial ring , where F is a field, says that for , with , that there exists unique polynomials

where where either or . We want to examine more closely the case where the

remainder .

Definition 5.1: Let F be a field and , with . We say that divides (or is a factor) of ) and write , if for some .

For example, since in since Also, every constant multiple of divides . For example,

divides since .

Facts1. If , the for every non-zero .

Proof:

2. Every divisor of a polynomial has degree less than or equal to .Recall that the of two positive integers a and b is the largest positive integer that divides a and b evenly. To find the for two polynomials

, we look for the polynomial that divides both

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and of highest degree. However, since, as we saw from Fact 1 above that if , the for every non-zero , we require the greatest common

divisor to be monic, that is, we require its leading coefficient to be 1. This assures that the greatest common divisor of two polynomials is unique. This is summarized in the following definition.

Definition 5.2: Let F be a field and , where or . The greatest common divisor of and , denoted by , is the monic polynomial of highest degree that both divides both and . In mathematical notation, we say that if is monic and (i) and .(ii) If and , then .

One method for finding the greatest divisor of two polynomials is to multiply the irreducible factors they have in common. We demonstrate with the following example.

Example 1: Find the greatest common divisor of the two polynomials and in

Solution: It can be verified that these polynomials have the following factorizations:

.

The common factors of both polynomials are

Note that the is factored to satisfy the requirement that the greatest common divisor is require to be monic. Hence,

.

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The Euclidean Algorithm for Polynomials in F[x]

To more efficiently find the greatest common of two polynomials, we can use the Euclidean Algorithm. The algorithm works similarly to how it works over the integers. Note that the last non-zero remainder may not be monic. However, by the greatest

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common divisor can be found simplify by factoring the constant from the leading coefficient of the last non-zero remainder. We state the algorithm as follows:

The Euclidean Algorithm

The Euclidean Algorithm makes repeated use of the division algorithm to find the greatest common divisor of two polynomials. If we are given two polynomials in

where , then if , then , where is the monic polynomial obtained by factoring the

leading coefficient of . If , then we compute

The , where is the monic polynomial obtained by factoring the leading coefficient of the last non-zero remainder .

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Example 2: Use the Euclidean Algorithm to find the greatest common divisor of the two polynomials and in .

Solution:

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Example 3: Use the Euclidean Algorithm to find the greatest common divisor of the two polynomials and in .

Solution:

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Theorem 5.3: For a field F, for any two polynomials , there are polynomials and where

Fact: When executing the Euclidean algorithm equations,

We can create a table to determine the , , and as follows:

Notes1. The quotients under Q and remainders under R are computed using the basic Euclidean

algorithm process. The table is complete when . The last non-zero remainder is used to find the greatest common divisor of and by converting , to a monic polynomial by factoring its leading coefficient. This monic

polynomial is the greatest common divisor, that is .2. The under U and under V are found using the formulas

.

3. For row i, we have . The values u and v where are found in the last row where

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.

The polynomials and are found by multiplying both sides of this equation by the multiplicative inverse of the leading coefficient of to obtain a monic polynomial for the greatest common divisor.

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Example 4: Use an Euclidean Algorithm table to find polynomials and where for the two polynomials

and in .

Solution:

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Example 5: Use an Euclidean Algorithm table to find polynomials and where for the two polynomials and

in .

Solution: First, we run the Euclidean Algorithm to show that using the following process: (note that the result is a monic polynomial)

Setting and and and using the equations

gives the following calculation for each row.,

.

, .

, .

The previous results give the following Euclidean Algorithm Table:

continued on next pageFrom the last row, we see that and This answer can be verified by checking

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.

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The following is a corollary resulting from the above results.

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Corollary 5.4: Let F be a field and , not both zero. A monic polynomial is the greatest common divisor of and if and only if

satisfies these conditions.:(i) and .(ii) if and , then .

Proof: Suppose is the greatest common divisor of and . By definition of the greatest common divisor, and . Also, we know there exists

such that

*

Now, if if and , then, by definition of divisibility, there exist polynomials where

and **

Substituting the results of ** into * gives

or

Hence, . Suppose satisfies conditions (i) and (ii). By condition (i), is a common

divisor of both and . Condition (ii) says must be the divisor of highest degree since if , then . Thus, is the greatest common divisor of and .

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Note: Two polynomials and are said to be relatively prime if .

Theorem 5.5: Let F be a field and . If and

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and are relatively prime, then .

Proof:

█Exercises

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1. Use the Euclidean Algorithm to find the greatest common divisor of the given polynomials.a. and in .b. and in .c. and in .d. and in .e. and in .f. and in .

2. For each exercise for Exercise 1, assign a and b and generate an Euclidean algorithm table to find polynomials and where .

3. Find in .

4. For a field F, let and suppose . If

and , prove that .

5. For a field F, let and suppose . If , prove that .

6. For a field F, let and suppose . Prove that.

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