Section 1: Rings and Fields - Radford Universitynpsigmon/.../Math623/mword/originals… · Web...
Transcript of Section 1: Rings and Fields - Radford Universitynpsigmon/.../Math623/mword/originals… · Web...
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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