Download Coursework 6

MTH4104
Introduction to Algebra
Assignment 6
Due on ***1 March 2017***
You are to write up for submission, in your groups, a set of careful and
professionally presented solutions to the starred questions.
The remaining questions are there for additional practice. It’s important
to work through lots of questions—remember that mathematics is not a
spectator sport!
1 Write out addition and multiplication tables for Z5 , like the ones for Z4 in the lecture
notes.
That is, let S := {0, 1, 2, 3, 4} be the canonical set of representatives for Z5 . For
each pair of elements a, b ∈ S, your tables should give the integers c, d ∈ S such that
[a]5 + [b]5 = [c]5
2
and
[a]5 [b]5 = [d]5 .
(a) Explain why [76]103 has a multiplicative inverse in Z103 .
(b) Find a non-negative integer b < 103 such that [76]−1
103 = [b]103 .
3 Which elements x ∈ Z11 satisfy the equation
[6]11 x + [1]11 =
[10]11
?
[6]11
Justify your answer.
* 4 Find X,Y ∈ Z7 that satisfy the simultaneous system of linear equations
[2]7 X + [5]7 Y = [1]7
[3]7 X +
Y = [2]7 .
5 Without using a calculator, write [280 ]19 as a class [a]19 where 0 ≤ a < 19. Try to be
efficient in your calculations.
6 How many of the elements of Z12 have multiplicative inverses? What about Z30 ?
Describe a general method that would allow you to calculate how many elements
of Zm have multiplicative inverses, without simply counting them all.
* 7 This question compares a naı̈ve way to take the “sum” and “product” of two sets of
integers to the definitions that we actually use in modular arithmetic.
(a) Prove that, for any integer m > 0, if X and Y are congruence classes of ≡m , then
the set
{x + y : x ∈ X, y ∈ Y }
is their sum X +Y within Zm .
(b) Give an example of an integer m > 0 and two congruence classes X,Y of ≡m such
that the set
{xy : x ∈ X, y ∈ Y }
is not the product XY within Zm .
Write down a general statement about how the above set is related to XY .
8
(a) Prove that, for any integer a,
[a]6 = [a]2 ∩ [a]3
is a true equation of sets.
(b) Let m and n be positive integers. Is there always an integer ` that makes the
equation of sets
[a]` = [a]m ∩ [a]n
true for any value of a? How do you find ` in terms of m and n?
9 Let F be the set {a + bI : a, b ∈ Z3 }, where I is a formal symbol. Define operations
of addition and multiplication on F by
(a + bI) + (c + dI) = (a + c) + (b + d)I,
(a + bI) · (c + dI) = (ac − bd) + (ad + bc)I.
That is, I is meant to behave like a square root of [−1]3 .
(a) How many elements does F have?
(b) Prove the distributive law in F.
(c) Prove the multiplicative inverse law in F.
[In fact, F is a field. I didn’t want to make you prove all the axioms, as that would
be too much to write. The multiplicative inverse law, part (c), is the most “interesting”
one.]