#15 on the book. b #15 on the book | bartlebyThomas W. Hungerford - Abstrac XO File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdfA' Read aloud V DrawF HighlightO Erase117of 621Capyr 2012 e La AI i a A May aot be A nd or hle or in pert D te edoie d, ed perty eom Ony bepmed mte Bodk endarChee. Hdtalnd tded tay ed co d t ty ate oad gmpat C Lag right o eddenal co g tit dt re i4.2 Divisibility in F[x]95Ceurestu(b) Give an example in Z[x] to show that part (a) may be false if the leadingcoefficient of g(x) is not a unit. [Hint: Exercise 5(b) with Z in place of Q.]15. Let R be a commutative ring with identity and a e R.(a) If a² = 0g, show that 1g + ax is a unit in R[x]. [Hint: Consider I – ax +(b) If a = OR, show that 1g + ax is a unit in R[x].16. Let R be a commutative ring with identity and a E R. If 1g + ax is a unit inR[x], show that d = Og for some integer n> 0. [Hint: Suppose that the inverseof 1g+ ax is bo + bjx + bzx² +...+ bx. Since their product is 1g, bo = 1r(Why?) and the other coefficients are all 0g-]17. Let R be an integral domain. Assume that the Division Algorithm alwaysholds in R[x). Prove that R is a field.18. Let p:R[x] R be the function that maps each polynomial in R[x] onto itsconstant term (an element of R). Show that o is a surjective homomorphismof rings.19. Let ç:Z[x] →Z_[x] be the function that maps the polynomial a, + ajx + . ..+ax in Z[x] onto the polynomial [ao] + [aj]x + • . · + [a]x*, where [a] denotesthe class of the integer a in Z,. Show that o is a surjective homomorphism ofrings.20. Let D:R[x]–→ R[x] be the derivative map defined byD(a, + ax + az + qx) = a, + 2a,x + 3a,x +...+ na,-!.Is Da homomorphism of rings? An isomorphism?C.21. Let h:R-→ S be a homomorphism of rings and define a function h:R[x] -> S[x]by the ruleh(a, + ajx + ... + apx") = h(as) + h(aj)x + h(az)x² + ... + h(a,)x".Prove that(a) h is a homomorphism of rings.(b) h is injective if and only if h is injective.(c) A is surjective if and only if h is surjective.(d) If R = S, then R[x] = S[x].22. Let R be a commutative ring and let k(x) be a fixed polynomial in R[x]. Provethat there exists a unique homomorphism ç:R[x] → R[x] such thato(r) = r for all r eRandp(x) = k(x).11:39 AMe Type here to searchEPICAiEPIC99+10/30/2020

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Math

Advanced Math

5. Let R be a commutative ring with identity and a e R. O a- O show that + ax is a unit in R[3), [HnE: Consider || – a (O) Fa'- On, show that, + axisa unit in R(3). 6. Let R be a commutative ring with identity and a e R. If l + ax is a unit in R(x), show that a - Og for some integer n>0. [Hint: Suppose that the inverse of la+ ax is by + bx + byx +...+ b. Since their product is l, bo- la (Why?) and the other coefficients are all O)

5. Let R be a commutative ring with identity and a e R. O a- O show that + ax is a unit in R[3), [HnE: Consider || – a (O) Fa'- On, show that, + axisa unit in R(3). 6. Let R be a commutative ring with identity and a e R. If l + ax is a unit in R(x), show that a - Og for some integer n>0. [Hint: Suppose that the inverse of la+ ax is by + bx + byx +...+ b. Since their product is l, bo- la (Why?) and the other coefficients are all O)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Kentville, a community of 10,000 people, resides next to a krypton mine, and there is a concern that the emission from the krypton smelter have resulted in adverse effects. Specifically, Kryptonosis seems to have killed 12 of Kentville’s inhabitants last year. A neighboring community, Lanesburg, has 25,000 inhabitants and is far enough from the smelter to not be affected by the emission. In Lanesburg, only three people last year died of Kryptonosis. Given that the number of deaths in Kentville and their causes last year were: Heart attack=7 Accidents=4 Kryptonosis=12 Other=6 What is the risk of dying of Kryptonosis in Kentville relative to non-contaminated locality?What is the risk of dying of Kryptonosis in Kentville relative to deaths due to other causes? How many times the chance of dying of Kryptonosis compared to dying of accidents ? How many times the chance of dying of Kryptonosis compared to Other causes?
From time to time, the UTM Human Resource (UTMHR) department observes various employees fortheir work productivity. Recently UTMHR wanted to check whether the four employees at theDepartment XYZ counters, serve on average the same number of customers per hour. The HR managerobserved each of the four employees for a certain number of hours. The following Table 5 gives thenumber of customers served by the four employees during each of the observed hours. employee a employee b employee c employee d 19 14 11 24 21 16 14 19 26 14 21 21 24 13 13 26 18 17 16 20 employee a employee b employee c employee d mean 21.6 14.8 15.0 22.0 s 3.4 ? 3.8 ? c) Find variance between sample,d) Find variance within sample, e) Calculate the test statistic, F.f) Determine the numerator and denominator.g) Find the critical value of F at the 5% significance level, and test the claim that the mean numberof customers served per hour by each of these four employees is…
From time to time, the UTM Human Resource (UTMHR) department observes various employees fortheir work productivity. Recently UTMHR wanted to check whether the four employees at theDepartment XYZ counters, serve on average the same number of customers per hour. The HR managerobserved each of the four employees for a certain number of hours. The following Table 5 gives thenumber of customers served by the four employees during each of the observed hours. employee a employee b employee c employee d 19 14 11 24 21 16 14 19 26 14 21 21 24 13 13 26 18 17 16 20 a) Define the hypothesis null, H0 and hypothesis alternative, H1.b) Given below are means and standard deviation for some of the employees. Calculate thestandard deviation for Employee B and D.
A research center claims that at least 31% of adults in a certain country think that their taxes will be audited. In a random sample of 800 adults in that country in a recent year, 26% say they are concerned that their taxes will be audited. At α=0.10, is there enough evidence to reject the center's claim? Complete parts (a) through (e) below. (a) Identify the claim and state H0 and Ha. Identify the claim in this scenario. Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or a decimal. Do not round.) A. The percentage of adults in the country who think that their taxes will be audited is not enter your response here%. B. enter your response here% of adults in the country think that their taxes will be audited. C. Less than enter your response here% of adults in the country think that their taxes will be audited. D. At least enter your response here% of adults in the country think…
A research center claims that at least 28% of adults in a certain country think that their taxes will be audited. In a random sample of1000 adults in that country in a recent year, 23%say they are concerned that their taxes will be audited. At α=0.10, is there enough evidence to reject the center's claim? Complete parts (a) through (e) below. (a) Identify the claim and state H0 and Ha. Identify the claim in this scenario. Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or a decimal. Do not round.) A. nothing% of adults in the country think that their taxes will be audited. B. Less than nothing% of adults in the country think that their taxes will be audited. C. At least nothing% of adults in the country think that their taxes will be audited. D. The percentage of adults in the country who think that their taxes will be audited is not nothing%. Let p be the population proportion of…
A research center claims that at least 31% of adults in a certain country think that their taxes will be audited. In a random sample of 800 adults in that country in a recent year, 28% say they are concerned that their taxes will be audited. At α=0.10, is there enough evidence to reject the center's claim? Complete parts (a) through (e) below. (a) Identify the claim and state H0 and Ha. Identify the claim in this scenario. Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or a decimal. Do not round.) A. ______% of adults in the country think that their taxes will be audited. B. The percentage of adults in the country who think that their taxes will be audited is not ______%. C. Less than ______% of adults in the country think that their taxes will be audited. D. At least ______% of adults in the country think that their taxes will be audited. (b) Find the critical value(s) and identify the rejection region(s). (c)…
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