Abstract Algebra 421
HW 4 抽象代数数学课业代写
You may use Subring Test Theorem 3.6 for your justifications in Section 3.1.
Problem (3.1.10). Is S = {(a, b) | a + b = 0} a subring of Z × Z? Justify your answer.
Problem (3.1.12). Let Z[i] denote the set {a + bi | a, b ∈ Z}. Show that Z[i] is a subring of C.
The following definition is needed for Exercises 41-43. Let R be a ring with identity.
If there is a smallest positive integer n such that n · 1R = 0R, then R is said to have characteristic n. If no such n exists, R is said to have characteristic zero.
Problem (3.2.41).
(a) Show that Z has characteristic zero and Zn has characteristic n.
(b) What is the characteristic of Z4 × Z6?
Problem (3.2.43). Let R be a ring with identity of characteristic n > 0.
(a) Prove that na = 0R for every a ∈ R.
(b) If R is an integral domain, prove that n is prime.
HW 5 抽象代数数学课业代写
Problem (4.4.2). Find the remainder when f(x) is divided by g(x):
(a) f(x) = x10 + x8 and g(x) = x − 1 in Q[x].
(d) f(x) = 2x5 −3x4 +x3 + 2x+ 3 and g(x) = x−3 in Z5[x]. (Smallest nonnegative answer.)
Problem (4.4.4).
(a) For what value of k is x − 2 a factor of x4 − 5x3 + 5x2 + 3x + k in Q[x]?
(b) For what value of k is x − 1 a factor of x4 + 2x3 − 3x2 + kx + 1 in Z5[x]? (Smallest nonnegative answer.)
Problem (4.4.8). Determine if the given polynomial is irreducible:
(d) 2x3 + x2 + 2x + 2 in Z5[x].
(f) x4 + x2 + 1 in Z3[x]
Problem (4.4.16). Let f(x), g(x) ∈ F[x] have degree ≤ n and let c0, c1, . . . , cn be distinct elements of F. If f(ci) = g(ci) for i = 0, 1, . . . , n, prove that f(x) = g(x) in F[x].
Problem (4.5.1). 抽象代数数学课业代写
Use the Rational Root Test to write each polynomial as a product of irreducible polynomials in Q[x]:
(a) −x4 + x3 + x2 + x + 2
(b) x5 + 4x4 + x3 − x2
Problem (4.5.5). Use Eisenstein’s Criterion to show that each polynomial is irreducible in Q[x]:
(a) x5 − 4x + 22
(b) 10 − 15x + 25x2 − 7x4
(c) 5x11 − 6x4 + 12x3 + 36x + 6
Problem (4.6.1). Find all the roots in C of each polynomial (one root is already given):
(a) x4 − 3x3 + x2 + 7x − 30; root 1 − 2i.
Problem (4.6.2). Find a polynomial in R[x] that satisfies the given conditions:
(a) Monic of degree 3 with 2 and 3 + i as roots
(b) Monic of least possible degree with 1 − i and 2i as roots
HW 6 抽象代数数学课业代写
Note: R denotes a ring and F denotes a field and p denotes a positive prime number.
Problem (4.1.5). Find polynomials q(x) and r(x) such that f(x) = g(x)q(x) + r(x), and r(x) = 0 or deg r(x) < deg g(x):
(d) f(x) = 4x4 + 2x3 + 6x2 + 4x + 5 and g(x) = 3x2 + 2 in Z7[x].
(The typography for the Division Algorithm is complex, so just tell me the remainder at each step. For example, if f(x) = x3 + x + 5 and g(x) = x + 1 in Q[x], the first remainder would be −x2 + x + 5. The second remainder would be 2x + 5, and so on. Of course give the final q(x) and r(x).)
Problem (4.1.13). Let R be a commutative ring. Let f(x) = a0 + a1x + a2x2 + · · · + anxn ∈ R[x]. If an ≠ 0R and f(x) is a zero divisor in R[x], prove that an is a zero divisor in R.
Problem (4.1.17). Let R be an integral domain. Assume that the Division Algorithm always holds in R[x]. Prove that R is a field.
Problem (4.2.5 & 6). 抽象代数数学课业代写
(Read the fairly long intro to Exercise 5 in the book.) #5 Use the Euclidean Algorithm to find the gcd of the given polynomials.
#6 Express each of the gcd’s in Exercise 5 as a linear combination of the two polynomials.
(c) x4 + 3x3 + 2x + 4 and x2 − 1 in Z5[x].
Problem (4.2.14). Let f(x), g(x), h(x) ∈ F[x], with f(x) and g(x) relatively prime. If f(x) | h(x) and g(x) | h(x), prove that f(x)g(x) | h(x).
Problem (4.3.12). Express x4 − 4 as a product of irreducibles in Q[x], in R[x], and in C[x].
Problem (4.3.15). (a) By counting products of the form (x + a)(x + b), show that there are exactly (p2 + p)/2 monic polynomials of degree 2 that are not irreducible in Zp[x].
(b) Show that there are exactly (p2 − p)/2 monic irreducible polynomials of degree 2 in Zp[x].
更多代写：cs北美网课代考推荐 托福枪手 英国艺术类网课代修 北美HIS代写 北美Creative Thinking代写 实分析数学作业代写
