Theorem 11.9. Suppose f(x) is a polynomial of positive degree in Z[x] and p is a...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Theorem 11.9. Suppose f(x) is a polynomial of positive degree in Z[x] and p is a prime number that does not divide the highest-degree coefficient of f(x). If the reduction [f](x) of f(x) modulo p is irreducible in Fp[x], then f(x) does not factor in Z[x] as a product of lower-degree polynomials. Exercise 11.16. Prove Theorem 11.9, using Theorem 11.8. Theorem 11.9 provides us with a new technique for proving that certain polynomials f(x) in Z[x] do not factor as a product of lower-degree polynomi- als: Find a prime number p that does not divide the highest-degree coefficient of f(x) (there are infinitely many from which to choose!) for which the reduc- tion [f](r) is irreducible in Fp[r]. The attraction of this technique resides in the fact that we can always determine, by the brute-force method of testing every possibility, whether a particular polynomial in F,[r] is irreducible. In particular, the irreducibility [f](x) in F,[x] may be easier to prove than the nonfactorizability of f(x) in Z[x] as a product of lower-degree polynomials. Let us consider two examples. Theorem 11.9. Suppose f(x) is a polynomial of positive degree in Z[x] and p is a prime number that does not divide the highest-degree coefficient of f(x). If the reduction [f](x) of f(x) modulo p is irreducible in Fp[x], then f(x) does not factor in Z[x] as a product of lower-degree polynomials. Exercise 11.16. Prove Theorem 11.9, using Theorem 11.8. Theorem 11.9 provides us with a new technique for proving that certain polynomials f(x) in Z[x] do not factor as a product of lower-degree polynomi- als: Find a prime number p that does not divide the highest-degree coefficient of f(x) (there are infinitely many from which to choose!) for which the reduc- tion [f](r) is irreducible in Fp[r]. The attraction of this technique resides in the fact that we can always determine, by the brute-force method of testing every possibility, whether a particular polynomial in F,[r] is irreducible. In particular, the irreducibility [f](x) in F,[x] may be easier to prove than the nonfactorizability of f(x) in Z[x] as a product of lower-degree polynomials. Let us consider two examples.
Expert Answer:
Answer rating: 100% (QA)
To prove Theorem 119 we will use Theorem 118 which states that if fx is a polynomial of positive degree in Zx and p is a prime number that does not di... View the full answer
Related Book For
Posted Date:
Students also viewed these accounting questions
-
Does there exist a polynomial of positive degree in Zo[z] that is a unit?
-
Planning is one of the most important management functions in any business. A front office managers first step in planning should involve determine the departments goals. Planning also includes...
-
The Chebyshevs polynomials are defined by Tn(x) = cos (n arcos x) n = 0, 1, 2, 3,, . (a) What are the domain and range of these functions? (b) We know that T0(x) = 1 and T1 (x) = x. Express T2...
-
A room is 6 m by 5 m by 3 m. (a) If the air pressure in the room is 1 atm and the temperature is 300 K, find the number of moles of air in the room. (b) If the temperature rises by 5 K and the...
-
The following 2010 information is available concerning the Drake Company, which adjusts and closes its accounts every December 31: 1. Salaries accrued but unpaid total $2,840 on December 31, 2010. 2....
-
Poplar Incorporated's management is interested in predicting travel and entertainment expense based on the number of expected sales calls on customers. Over the past 50 weeks, the company's Sales...
-
List the problems with life-cycle assessment. Explain which level of management is appropriate for addressing each of these problems.
-
Presented below are the financial statements of Newman Company. Additional data: 1. Dividends declared and paid were $27,000. 2. During the year equipment was sold for $8,500 cash. This equipment...
-
If the Atlanta Hawks have consistent yearly profits of $12 million, how much would you be willing to pay for this team, given a 4% interest rate. A) $11,538,462 B) $240,000,000 C) $12,489,729 D)...
-
Consider the process model in Fig. 7.14. This model captures a simplified process for handling applications for mortgages. There are two checks involved. CT1 deals with a check of the financial...
-
find the zeros f ( x ) = 2 7 x ^ 3 - 6 4
-
True or False: Trial weights can be attached to the outer edge of a fan wheel just using the clamping bolt.
-
What are the two most important factors when preparing to impact a machine or part for natural frequency identification?
-
True or False: The RMS can be calculated for a nonperiodic time waveform.
-
True or False: A successful impact analysis of a motor end bracket can be accomplished with the rotor at rest in sleeve bearings.
-
What is the critical speed map good for?
-
A point charge Q1 = 6.3 nC is at the point (0.309 m, 0 m); a charge Q2 = -1.61 nC is at 0 m, 0.25 m). and a charge Q3 = 5.74 nC is at (0 m, 0 m). What is the magnitude of the net force on Q3 due to...
-
You have just begun your summer internship at Omni Instruments. The company supplies sterilized surgical instruments for physicians. To expand sales, Omni is considering paying a commission to its...
-
Repeat Exercise 7 using the Cubic Spline Algorithm. Repeat exercise 7
-
Repeat Exercise 8 using the Gauss-Seidel method. Repeat exercise
-
The data for Exercise 5 were generated using the following functions. Use the error formula to find a bound for the error, and compare the bound to the actual error for the cases n = 1 and n = 2. a....
-
The man has a mass of 78 kg and stands motionless at the end of the diving board. If the board has the cross section shown, determine the maximum normal strain developed in the board. The modulus of...
-
The steel rod having a diameter of 1 in. is subjected to an internal moment of M = 300lbft. Determine the stress created at points A and B. Also, sketch a three-dimensional view of the stress...
-
The steel beam has the cross-sectional area shown. If w=5 kip/ft, determine the absolute maximum bending stress in the beam. W 8 8 ft- 8 ft W ft- -8 ft- .8 in. 0.30 in. 0.3 in.- 10 in. 0.30 in.
Study smarter with the SolutionInn App