(e) +x-1 in Ca] 2. Prove that every nonzem f(x) e Flx] has a unique monie asnciate in F1x. 3 List all asociater of (O P+x+1inZ{ 4. Show that a nonzero polynomial in Z has exactly p- 1 asociates.
(e) +x-1 in Ca] 2. Prove that every nonzem f(x) e Flx] has a unique monie asnciate in F1x. 3 List all asociater of (O P+x+1inZ{ 4. Show that a nonzero polynomial in Z has exactly p- 1 asociates.
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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
#3 on the picture
![b A MATLAB program that creates
POE Thomas W. Hungerford - Abstrac X
O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf
+
A' Read aloud V Draw
F Highlight
O Erase
125
of 621
4.13 shows that p1(x)|q;(x) for somej. After rearranging and relabel-
ing the q(x)'s if necessary, we may assume that pi(x)|q1(x). Since q1(x)
is irreducible, p(x) must be either a constant or an associate of q(x).
However, pi(x) is irreducible, and so it is not a constant. Therefore, pi(x)
is an associate of q1(x), with p1(x) = q9(x) for some constant c1. Thus
91(x)[c,P2(x)P;(x) • .• p(x)] = P(x)P{x) • .· p{x) = 9(x)4(x)•· · · q£x).
Canceling q,(x) on each end, we have
P2(x)[cP3(x) • · • p,(x)] = 9{x)q;(x) •.• q&x).
Complete the argument by adapting the proof of Theorem 1.8 to F[x],
replacing statements about t, with statements about associates of
9,(x). I
I Exercises
NOTE: F denotes a field and p a positive prime integer.
A. 1. Find a monic associate of
(a) 3x + 2x2 + x + 5 in Q[x]
(b) 3x – 4x2 + 1 in Zgx]
(c) ix +x – 1 in C[x]
2. Prove that every nonzem f(x) E Fx] has a unique monic associate in F[x].
3. List all associates of
(a) x² + x + 1 in Z{x]
(b) 3x + 2 in Z,[x]
4. Show that a nonzero polynomial in Z(x] has exactly p – 1 associates.
5. Prove that f(x) and g(x) are associates in F[x] if and only if f(x)|g(x) and
g(x)[f(x).
6. Show that x +1 is irreducible in Q[x]. [Hint: If not, it must factor as
(ax + b)(cx + d) with a, b, c, d e Q; show that this is impossible.]
7. Prove that f(x) is irreducible in F[x] if and only if each of its associates is
irreducible.
Opte 2012Can lang A Rig Rat May aot be opd med or dptic e ae or in pat D so deais d trd party eonte may be
fnteBodk a Btalvt
ded thet ny pnd t d ot aty dha be oat ingpeias Ong Laing righto dddonl cot theit gta to in
104 Chapter 4 Arithmatic in F[x]
8. If f(x) E F[x] can be written as the product of two polynomials of lower
degree, prove that f(x) is reducible in F[x]. (This is the second part of the
proof of Theorem 4.11.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27260fae-539c-4ca6-9fed-6022b8026087%2F1dc14dcb-eb74-40c6-a315-6b3c9bcd9842%2Fndxuuab_processed.png&w=3840&q=75)
Transcribed Image Text:b A MATLAB program that creates
POE Thomas W. Hungerford - Abstrac X
O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf
+
A' Read aloud V Draw
F Highlight
O Erase
125
of 621
4.13 shows that p1(x)|q;(x) for somej. After rearranging and relabel-
ing the q(x)'s if necessary, we may assume that pi(x)|q1(x). Since q1(x)
is irreducible, p(x) must be either a constant or an associate of q(x).
However, pi(x) is irreducible, and so it is not a constant. Therefore, pi(x)
is an associate of q1(x), with p1(x) = q9(x) for some constant c1. Thus
91(x)[c,P2(x)P;(x) • .• p(x)] = P(x)P{x) • .· p{x) = 9(x)4(x)•· · · q£x).
Canceling q,(x) on each end, we have
P2(x)[cP3(x) • · • p,(x)] = 9{x)q;(x) •.• q&x).
Complete the argument by adapting the proof of Theorem 1.8 to F[x],
replacing statements about t, with statements about associates of
9,(x). I
I Exercises
NOTE: F denotes a field and p a positive prime integer.
A. 1. Find a monic associate of
(a) 3x + 2x2 + x + 5 in Q[x]
(b) 3x – 4x2 + 1 in Zgx]
(c) ix +x – 1 in C[x]
2. Prove that every nonzem f(x) E Fx] has a unique monic associate in F[x].
3. List all associates of
(a) x² + x + 1 in Z{x]
(b) 3x + 2 in Z,[x]
4. Show that a nonzero polynomial in Z(x] has exactly p – 1 associates.
5. Prove that f(x) and g(x) are associates in F[x] if and only if f(x)|g(x) and
g(x)[f(x).
6. Show that x +1 is irreducible in Q[x]. [Hint: If not, it must factor as
(ax + b)(cx + d) with a, b, c, d e Q; show that this is impossible.]
7. Prove that f(x) is irreducible in F[x] if and only if each of its associates is
irreducible.
Opte 2012Can lang A Rig Rat May aot be opd med or dptic e ae or in pat D so deais d trd party eonte may be
fnteBodk a Btalvt
ded thet ny pnd t d ot aty dha be oat ingpeias Ong Laing righto dddonl cot theit gta to in
104 Chapter 4 Arithmatic in F[x]
8. If f(x) E F[x] can be written as the product of two polynomials of lower
degree, prove that f(x) is reducible in F[x]. (This is the second part of the
proof of Theorem 4.11.)
Expert Solution
Step 1
Given that both are field hence we can
Find corresponding unit of field then easily we can find its associates.
Step by step
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