1. The integers modulo n. Fix a non-zero integer n ∈ Z.1 Given any other integer m ∈ Z, the reduction of m modulo n is defined as the remainder of m when divided by n. The reduction of m modulo n is written as m (mod n), and takes values between 0 and n− 1 (inclusive). Consider the set
Z/nZ := {0, 1, 2, . . . ,n− 1}.
This set has two operations.
Addition: an operation +n : Z/nZ×Z/nZ → Z/nZ,
given on a,b ∈ Z/pZ by a +n b := a + b (mod n),
where the + on the right hand side is the usual addition in Z. Multiplication: an operation
·n : Z/nZ×Z/nZ → Z/nZ,
given a,b ∈ Z/pZ by a ·n b := a · b (mod n),
where the · on the right hand side is the usual multiplication in Z.2
(a) Write out the multiplication tables for Z/3Z and Z/4Z. That is, fill in the tables below with the product of the row and column label in each box. An example is given for Z/2Z.3
Table 1: Multiplication in Z/2Z
0 1
0 0 · 0 = 0 0 · 1 = 0
1 1 · 0 = 0 1 · 1 = 1
1Remember that the symbol Z represents the set of all whole numbers, i.e. {. . . ,−2,−1, 0, 1, 2, . . .}. 2It’s traditional to write addition and multiplication of integers modulo n without the subscripts when it is implicit that
everything is being reduced modulo n, but for this problem we will include all subscripts to distinguish ordinary addition/ multiplication from that in Z/nZ.
3You don’t need to write out the equation inside each box when you fill out the tables for Z/3Z and Z/4Z.
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115A – HW1 October 1, 2021
Table 2: Multiplication in Z/3Z
0 1 2
0
1
2
Table 3: Multiplication in Z/4Z
0 1 2 3
0
1
2
3
(b) Recall that a number p ∈ Z is called a prime number if the only positive divisors of p are 1 and p itself. Prove that Z/pZ is a field when p is prime.4
NOTE. (Added 10/1.) you may use the following basic facts from number theory, without proving them:
Fact 1. Given integers a,b, and n, the following equality holds:
(a + b) (mod n) = ( (a (mod n)) + (b (mod n))
) (mod n).
(a · b) (mod n) = ( (a (mod n)) · (b (mod n))
) (mod n).
Fact 2. Given nonzero integers a and b, gcd(a,b) denote the greatest common divisor of a and b: more precisely, gcd(a,b) is an integer that divides both a and b, and if k is any other number that divides both a and b, then k ≤ gcd(a,b). It is always possible to write gcd(a,b) as a sum of integer multiples of a and b. In symbols: there exist integers x,y such that
xa + yb = gcd(a,b).
(c) A number n is called a composite number if there exist p,q such that n = pq, where p,q ∈ N are greater than 1. Show that Z/nZ is not a field when n is composite. Explain what goes wrong.5
(d) Give three examples of vector spaces over Z/2Z.
2. Is R a vector space over Q? If yes, is Q a vector subspace of R over Q? Justify your answers, but you do not need to provide a complete proof.
4If you are struggling with this problem, begin by identifying the additive and multiplicative units. Also see lecture notes from 10/1 for a complete proof in the case p = 3.
5Hint: look at the multiplication table for Z/4Z vs. Z/3Z.
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115A – HW1 October 1, 2021
3. General field-valued functions as vector spaces. Let S denote a set and F a field. We consider the set Fun(S,F) of all functions f : S → F. 6
(a) Define addition on elements of Fun(S,F).
(b) Define scalar multiplication of elements of Fun(S,F) by elements of F .
(c) Show that the two operations defined above make Fun(S,F) a vector space.
(d) Deduce that the set Fun(R,R) of all functions from R to R forms a vector space over R. (e) Recall that, in class, we claimed that the set C∞(R) of differentiable functions from R to R forms
a vector space over R. Prove this by verifying that C∞(R) is a vector subspace of Fun(R,R) over R.
4. Uniqueness of inverses. Using the Cancellation Law for Vector Addition (textbook Theorem 1.1, written in 9/27 class notes), prove that additive inverses are unique: given a vector space V over a field F and an element v ∈ V , there is a unique element w ∈ V such that v + w = 0.7
5. Linear combinations. Write the polynomial f(x) := x4 + 4×2 − √
2 as a linear combination of the polynomials
a(x) := x4 −x,
b(x) := x3 + x2,
c(x) := √
2×2,
d(x) := x− 1,
e(x) := 1
in the vector space P4(R) over R. More explicitly: find real numbers λ1,λ2,λ3,λ4,λ5 such that
f(x) = λ1a(x) + λ2b(x) + λ3c(x) + λ4d(x) + λ5e(x).
6. Textbook 1.2, # 2, 3
7. Textbook 1.2 # 13
8. Textbook 1.3 #5
9. Textbook 1.3 # 27
10. Textbook 1.4 #2(a),(b)
11. Textbook 1.4 # 6.
12. Additional recommended problems (not to be turned in). At some point, I suggest you look over the statement of these problems to see if you think you could solve them, and ask about them in TA session/ office hours if you think they look interesting or tricky. These can also be “guide problems” when you are studying for an exam and want more practice.
Textbook: Section 1.2, # 1, 8, 10, 11, 12, 14, 17, 22. Section 1.3, # 2, 4, 6, 7, 11, 12, 13, 30 Section 1.4, # 1, 4 (a)+(b), 6, 14.
6Recall that a function from S to F is an assignment to each element s ∈ S of a unique value f(s) ∈ F . In this case, we do not require our functions to have any special properties, i.e. continuity or differentiability, which a priori don’t make sense for a general set S and field F .
7If you need inspiration, see the proof that 0 is unique, given in the lecture 2 notes.
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