# Abstract algebra questions and answers Recent questions in Abstract algebra
Abstract algebra
ANSWERED ### If a, b are elements of a ring and m, n ∈ Z, show that (na) (mb) = (mn) (ab)

Abstract algebra
ANSWERED ### Let f1,…,fr be complex polynomials in the variables x1,…,xn let V be the variety of their common zeros, and let I be the ideal of the polynomial ring R=C[x1,…,xn] that they generate. Define a homomorphism from the quotient ring $$\displaystyle\overline{{{R}}}={R}&#{x}{2}{F}$$. I to the ring RR of continuous, complex-valued functions on V.

Abstract algebra
ANSWERED ### Write formulas for the isometries in terms of a complex variable z = x + iy.

Abstract algebra
ANSWERED ### Explain why (a) Z_9 is not isomorphic to Z_3 × Z_3. (b) Z_9 × Z_9 is not isomorphic to Z_9 × Z_3 × Z_3.

Abstract algebra
ANSWERED ### Let $$\displaystyle{n}≥{8}$$. Give a $$\displaystyle{b}{i}{g}-Θ$$ estimate for the number of circuits of length 8 in Kn. $$\displaystyleΘ{\left({n}\right)}Θ{\left({n}{8}\right)}Θ{\left({8}\right)}Θ{\left({8}{n}\right)}Θ{\left({8}{n}\right)}$$

Abstract algebra
ANSWERED ### The front wheel of a tricycle has a circumference of 63 ​inches, and the back wheels have a circumference of36 inches. If points P and Q are both touching the sidewalk when Felicity starts to​ ride, how far will she have ridden when P and Q first touch the sidewalk at the same time​ again?

Abstract algebra
ANSWERED ### If a, b are elements of a ring and m, $$n \in Z$$, show that $$(na) (mb) = (mn) (ab)$$

Abstract algebra
ANSWERED ### Let A be an uncountable set, B a countable subset of A, and C the complement of B in A. Prove that there exists a one to one correspondence between A and C.

Abstract algebra
ANSWERED ### The Ramirez family is putting a koi pond in their backyard. They need to estimate the area of the pond to know how many fish they can put in the pond. A scale drawing of the pond is shown below.In the drawing, each square represents one square foot. What do you know? What do you need to know?

Abstract algebra
ANSWERED ### Explain why (a) $$Z_9$$ is not isomorphic to $$Z_3 \times Z_3$$; (b) $$Z_9 \times Z_9$$ is not isomorphic to $$Z_9 \times Z_3 \times Z_3$$.

Abstract algebra
ANSWERED ### In Exercises 6 through 9, write the set in the form $$\{x|P(x)\}$$, where P(x) is a property that describes the elements of the set. 6. $$\{2, 4, 6, 8, 10\}$$ 7. $$\{a, e, i, o, u\}$$ 8. $$\{1, 8, 27, 64, 125\}$$ 9. $$\{-2, -1, 0, 1, 2\}$$

Abstract algebra
ANSWERED ### Question 4 (Module Outcome #4): Find the best-case, worst-case and average-case number of < comparisons are performed by the following piece of pseudocode. Precondition: $$n\in\{1,3,5,7,9\}\ \text{while}\ n < 6\ \text{do}\ n\leftarrow n+3$$

Abstract algebra
ANSWERED ### In the froup $$\displaystyle{Z}_{{12}}$$, find $$|a|, |b|$$, and $$|a+b|$$ $$a=5, b=4$$

Abstract algebra
ANSWERED ### Prove the following. (1) $$Z \times 5$$ is a cyclic group. (2) $$Z \times 8$$ is not a cyclic group.

Abstract algebra
ANSWERED ### Let H be a normal subgroup of a group G, and let $$m = (G : H)$$. Show that $$a^{m} \in H$$ for every $$a \in G$$

Abstract algebra
ANSWERED ### If U is a set, let $$\displaystyle{G}={\left\lbrace{X}{\mid}{X}\subseteq{U}\right\rbrace}$$. Show that G is an abelian group under the operation $$\oplus$$ defined by $$\displaystyle{X}\oplus{Y}={\left({\frac{{{x}}}{{{y}}}}\right)}\cup{\left({\frac{{{y}}}{{{x}}}}\right)}$$

Abstract algebra
ANSWERED ### Let (Z,+) be a group of integers and (E,+) be a group of even integers. Find and prove if there exist an isomorphism between them.

Abstract algebra
ANSWERED ### Let F be a field, and $$\displaystyle{p}{\left({x}\right)}\in{F}{\left[{x}\right]}$$ an irreducible polynomial of degreed. Prove that every coset of $$\displaystyle{F}\frac{{{x}}}{{{p}}}$$ can be represented by unique polynomial of degree stroctly less than d. and moreover tha these are all distinct. Prove that if F has q elements, $$\displaystyle{F}\frac{{{x}}}{{{p}}}$$ has $$\displaystyle{q}^{{d}}$$ elements.

Abstract algebra
ANSWERED ANSWERED 