Abstract Algebra Answers

Jon walks 1850m to school. His friend Sam walks 1.5km to school. What is the difference between the distances the two boys walk?

Wood can be bought in 2m lengths. Steve is making a picture frame that is 45.5cm by 61cm. How many lengths of wood does he need to buy and how much will be left over?

a store stocks 50 birthday cards, 45 get well cards and 10 anniversary cards

what simplest form in a fraction is the get well cards

Mr. Fuller has paid $144 in interest on a loan of 1,800 borrowed for a year. What is the rate of interest on the loan?

A polynomial function P(x) has degree n. the graph in the standard (x,y) coordinate plane of y = P(x) contains exactly 3 points on the x-axis. Which of the following could NOT be the value of n?
6, 5, 4, 3, 2

Let Z be the ring of integers. Consider the function
f : Z[x] → Z
defined by f(g(x)) = g(0). for example, f(x2 + 1) = 1.
(a) Show that f is a ring homomorpism.
(b) What is the ker(f)?

Let R be a field. Prove that the only ideals in R are either {0} or R itself. (Hint:
Show that if the ideal is non-empty, then it must contain 1).

(a) Let z =1 + i. What is z100 in both rectangular and polar form?
(b) Find the two square roots of 1 + *Square root* 3 i. Express them in polar form.

A new cruise ship line has just launched 3 new ships: The pacific paradise, the Caribbean Paradise, and the Mediterranean Paradise. The caribbean paradise has 27 more deluxe staterooms than the Pacific Paradise.The Mediterranean Paradise has 32 fewer deluxe staterooms than four times the number of deluxe staterooms of the pacific paradise.Find the number of deluxe staterooms for each of the ships if the total number of deluxe staterooms for the three ships is 1153

Let R and S be rings. The direct product of R and S,
R ⊕ S = {(r, s) : r ∈ R, s ∈ S}
is a ring, where
(r1, s1) + (r2, s2) = (r1 + r2, s1 + s2)
(r1, s1) x (r2, s2) = (r1 x r2, s1 x s2)

(a) List the elements of Z2 ⊕ Z3 and Z3 ⊕ Z2.
(b) Are they structurally the same (that is isomorphic)? If so, how should an
element in Z2 ⊕ Z3 be identified in Z3 ⊕ Z2?