Practice Question Set: Ideals in Rings and Field Extensions

In this question you will practise working with ideals in rings that are extensions of degree 2 over the integers. It involves a number of concepts from Chapter 17.

Let R be the ring Z[

√

−19 ].

(a) Let A be a subset of R defined by

A = {s + t

√

−19 : s,t ∈ Z, s ≡ t(mod 5)}.

Show that A is an ideal in the ring R. [3]

(b) The subset

B = {u + v

√

−19 : u, v ∈ Z, u ≡ −v (mod 5)}

is also an ideal in the ring R (you do not need to prove this). Show that

AB = #5".

This question tests your ability to deal with field extensions. You will need to use concepts and results from Chapters 17 and 18.

Let u and v be positive integers, u '= 4v, such that none of u, v and uv is a perfect square. Under these assumptions, it can be shown that

√

u /∈ Q(

√

v)

and √

v /∈ Q(

√

u); you do not need to prove this. Further, let α = 2√

u −

√

v.

(a) Show that α

3 − (4u + 3v)α is a non-zero element of Q(

√

v). [2]

(b) With the help of part (a), or otherwise, show that Q(√

u, √

v) = Q(α). [3]

(c) Deduce that the degree [Q(α) : Q] is equal to 4. [2]

(d) Determine the minimal polynomial for α over Q. [3]

Hint: begin by evaluating α

2.

In this question you are asked to perform calculations inafinite field. It is based on concepts developed in Chapter 18. Consider the finite field F = Z2(ω), where ω is a root of the irreducible polynomial f(x) = x

4 + x

3 + 1 over Z2, so that ω

4 + ω

3 + 1 = 0; the order of

F is 2

4 = 16. By Proposition 1.7 of Chapter 18, every element of F can be written in a unique way in the form a+bω +cω

2 +dω

3

for some a, b, c, d ∈ Z2.

(a) Show that for every element z ∈ F, we have z + z = 0. [1]

(b) With the help of one of the methods explained in Worked Exercises 3.10 and 3.11 of Chapter 18, or otherwise, find the multiplicative inverse of the element 1 + ω + ω

Question 2

2

in F. [4]

(c) By evaluating ω

5 but no other powers of ω, show that ω is a primitive element of F. [3]

This question tests your skills in handling splitting fields and extensions of the field of rational numbers. You will need to use concepts and results from Chapters 17 and 18.

The polynomials f(x) = x

6 − 2 and g(x) = x

2 + x + 1 are both irreducible

over Q; you do not need to prove this. Let ε be a root of g.

(a) Verify that ε

3 = 1. [1]

(b) With the help of the elements √6 2 and ε, write down the set of all six distinct roots of the polynomial f. [2]

(c) Show that the degree of the splitting field K of f over Q is equal to 12, as follows.

(i) By considering the roots of f, show that both

√6

2 and ε lie in K. [1]

(ii) Show that the polynomial g is irreducible over the field Q(

√6

2). [2]

(iii) Using the results obtained in parts (c)(i) and (c)(ii), deduce that [K : Q]=12 with the help of the KLM theorem. 

This question is designed to test your understanding of (im)possibility of construction with the help of ruler and compasses. It builds on the material of Chapter 19.

Duplication of a cube and quadrature ofacircle are two classical problems of Greek geometry. Let us consider their analogues, namely the problems of ‘triplicating a cube’ and ‘cubingaball’. Triplicating a cube means constructing a cube of three times the volume ofagiven cube, and cubing a ball means constructing a cube of the same volume asagiven ball; in both cases, ‘constructing’ refers to using ruler and compasses only. We may also assume that the given cube has unit side length and the given ball has unit radius.

(a) Show thataunit cube cannot be triplicated by a construction using ruler and compasses only, following the steps below.

(i) Show that ifaunit cube can be triplicated using ruler and compasses only, then u =

√3

3 is a constructible number. [2]

(ii) Find the minimal polynomial for u over Q, and apply Corollary 2.8 of Chapter 19. [3]

(b) Show that a unit ball cannot be cubed by a construction using ruler and compasses only, following the steps below.

(i) Show that ifaunit ball can be cubed using ruler and compasses only, then v = !3 4π/3 is a constructible number. [2]

Question 3

(ii) Show that v is transcendental over Q, and apply Theorem 2.7 of Chapter 19. [3]

This question tests your knowledge of encryption and decryption using the RSA method. The numbers involved are deliberately chosen to be small enough to focus on your understanding without excessive calculations. Alice and Bob decide to use an RSA cryptosystem with public key (77, 13) for communication.

(a) Alice wants to send Bob the message m = 29. Determine Alice’s ciphertext c. [4]

(b) Determine Bob’s private key. [2]

(c) Carry out Bob’s decryption of Alice’s ciphertext c, and compare his result with Alice’s message.

This question can be done at any point during your study of Book E. An important part of being a mathematician is being able to communicate mathematics using mathematical notation in different settings. For this TMA, we would like you to post something in the Book E forum that is about the mathematics in M303 and contains mathematical notation. There is a forum post explaining how to use mathematical notation in the Book E forum.

To earn full marks, provide either a screenshot of your forum post or directions as to where to find it (giving thread title, time and date). (If you are not inaposition to provide such evidence, or if you cannot use the forum for any reason, contact your tutor for advice.) [3]

Part B Formative

This question is designed to help you to practise techniques relevant to Question 1 in TMA 06. It concerns material covered in Chapter 21 of Book F.

Let d be the Euclidean metric on R 2 given by

d

"

(x1, y1),(x2, y2)

#

=

!

(x1 − x2)

2 + (y1 − y2)

2,

for (x1, y1),(x2, y2) ∈ R

2

.

Consider the following four sets in R

2

:

W = {(x, y) ∈ R

2

: y < π} ∪ {(x, y) ∈ R

2

: y > π and y ≤ 7},

X = {(x, y) ∈ R

2

: x = y} ∪ {(3, y) ∈ R

2

: y ∈ R},

Y = {(q, π) ∈ R

2

: q ∈ Q},

Z = {(x, y) ∈ R

2

: x

2 + (y − π)

2 = π

2

} ∪ {(x, y) : x

2 + (y + π)

2 < π

2

}.

For each set, do the following.

(a) On a separate diagram, sketch the set, by hand or otherwise. (If you are unsure how to represent something on a diagram, or are unable to sketch it, then describe what you mean in words. You are also free to use a combination of sketching and writing.) [4]

Question 4

(b) Write down whether the set is d-connected or d-disconnected. If the set is d-disconnected, then write down a d-disconnection. [6]

Hint: when using the Euclidean metric, it is always worth guessing. 

This question enables you to practise your calculation skills in rings of the form Z[

√−n ]. It involves a number of concepts from Chapter 17 and also the concepts of prime and irreducible elements of a ring, introduced and studied earlier in Section 2, and specifically Subsection 2.2, of Chapter 12.

Let n ≥ 3 be an odd positive integer that is not a perfect square, and let

R = Z[

√

−n ], with multiplicative norm N(r + s

√

−n) = r

2 + ns2

for every

element r + s

√

−n ∈ R. Let p ∈ Z, p ≥ 2, be a prime in Z such that p divides n + 1.

(a) By finding distinct factorisations of n + 1 in R, show that p is not a prime in R. [3]

Hint: see the solution to Exercise 2.13(b) of Chapter 17.

(b) Show that there is no element r + s

√

−n ∈ R, for r, s ∈ Z, with norm

equal to p. [3]

Hint: determine first which of the two numbers p, n is smaller than the other.

(c) With the help of the result of part (b), or otherwise, show that p is an irreducible element of R. [4]

This question is designed to develop your understanding of the material in Section 2 of Chapter 18 on finite fields. It will help you to practise and develop your skills in handling extensions of the field Q of rational numbers by adjoining algebraic elements.

(a) Determine the minimal polynomials for √3

7 and √4

7 over Q, and

determine the degrees [Q(

√3

7) : Q] and [Q(

√4

7) : Q]. [3]

(b) By considering common factors of the degrees found in part (a), or otherwise, show that

√3

7 ∈/ Q(

√4

7). [2]

(c) Hence determine the degree [Q(

√3

7,

√4

7) : Q]. [1]

(d) Show that Q(

√3

7,

√4

7) = Q(

√3

7 ·

√4

7) = Q(

12√

7). [2]

Hint: express the various roots of 7 as powers of 71/12

This question is designed to develop your understanding of the material in Section 3 of Chapter 18 on fields and their extensions. It will help you to practise and develop your skills in making calculations in finite fields, including determining multiplicative inverses and primitive elements. Consider the finite field F = Z5(ω), where ω

Question 5A (Constructibility)

2 = −3 − 2ω = 2 + 3ω; the order

of F is 52 = 25. Proposition 1.7 of Chapter 18 tells us that every element of F can be written uniquely in the form a + bω for some a, b ∈ Z5. By Theorem 3.3 of Chapter 18, the multiplicative group of F is cyclic, of order 5

2 − 1=24. In particular, if ξ is a primitive element of F (Definition 3.4 of Chapter 18), then every non-zero element of F is equal to ξ

i

for some i such that 0 ≤ i ≤ 23.

(a) Determine ω

3 and ω

6

in the form a + bω for suitable a, b ∈ Z5. [2]

(b) Show that ω is a primitive element of Z5(ω). [3]

(c) By one of the methods explained in Worked Exercises 3.10 and 3.11 of Chapter 18, or otherwise, determine the multiplicative inverse ω −1 of ω in the form ω −1 = a + bω for suitable a, b ∈ Z5. [2]

(d) With the help of Corollary 3.5 of Chapter 18, determine all primitive elements of Z5(ω) in the form a + bω for suitable a, b ∈ Z5. [4]

This question is designed to develop your understanding of (im)possibility of construction with the help of ruler and compasses. It builds on the material of Chapter 19.

(a) Assume that you have a ruler with marks at 0, 1 and √6

2. Determine if it is possible to square a circle of radius

√

2 with such a ruler and compasses. [4]

(b) Assume that you have a ruler with marks at 0, 1 and π 2

. Determine if it is possible to square a circle of radius √

2 with such a ruler and compasses. [3]

(c) Determine if it is possible to duplicate a unit cube with compasses and a ruler with marks at 0, 1 and √3 4. [2]

(d) Triplicating a unit cube means constructing a side ofacube of volume 3. Determine if it is possible to triplicate a unit cube with compasses and a ruler with marks at 0, 1 and √6 3. [2]

This question gives you the opportunity to develop your knowledge of encryption and decryption in cryptosystems based on elliptic curves over finite fields, covered in Chapter 20. The numbers involved are deliberately chosen to be small enough to focus on your understanding without excessive calculations.

Alice and Bob want to communicate using the Diffie–Hellman–ElGamal system based on the elliptic curve E over GF11 given by the equation y

2 = x

3 + 2x + 5, with the point P = (0, 4). Bob decided to have sB = 2 as his private key. By experimenting in the group (E, +), Alice found out that 5P = (4, 0).

(a) Using the above information, show that the order of P in the group (E, +) is equal to 10. [2]

Hint: determine first the order of the point (4, 0) by working out its inverse in (E, +).

(b) With the help of part (a) and the result of Theorem 2.20 of Chapter 20, determine the order of (E, +) and show that this group is cyclic, without listing any of the remaining elements. [3]

(c) Determine the point PB = sBP. [2]

(d) Alice wants to encrypt the message M = (0, 7) = −P ∈ E, choosing a random integer k = 3. Determine the points M1 and M2 that Alice sends to Bob. [3]

(e) Describe Bob’s decryption process for Alice’s messages M1 and M2. [1] Hint: calculations in parts (c) and (d) will simplify if you realise that by parts (a) and (b), every element of (E, +) is a multiple of P.