Elementary Number Theory - Lecture Notes 2020-2021

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Questions and Answers

Which operation forms a group in the set Zn according to the text?

Addition (correct)

Subtraction

Division

Multiplication

In which set does every element not have an inverse under multiplication?

Z6 (correct)

What property is required for an element in Zn to have a multiplicative inverse?

Having gcd(a, n) = 1 (correct)

Being even

Having a square root

Being a prime number

Which binary operation is defined on Zn by multiplication?

Exponentiation Signup and view all the answers

What does the division algorithm state?

There exist q, r ∈ Z such that m = nq + r and 0 ≤ r < n for m ∈ Z and n ∈ N Signup and view all the answers

What is the condition for an element in Z to have a multiplicative inverse according to the text?

$gcd(a, n) = 1$ Signup and view all the answers

What is the purpose of using induction in proving P(m) for m ≥ 0?

To demonstrate that P(m) holds true for all integers m Signup and view all the answers

In Z6, which elements do not have multiplicative inverses?

[2]6 and [4]6 only Signup and view all the answers

In the proof, what does it mean when it states 'Letting q = q′ + 1 and r = r′ proves P (m)'?

It demonstrates that P(m) holds true by substituting values of q and r Signup and view all the answers

What does the existence of b in Z imply according to Proposition 5.10?

$ab \equiv 1$ (mod n) Signup and view all the answers

What happens if r′ = 0 in the proof?

m = n(−q ′ ) and thus P(m) holds Signup and view all the answers

$ax \equiv 1$ (mod n) by Proposition 4.25 implies:

$ax+n(-y) = 1$ Signup and view all the answers

How does the proof handle cases when m < 0?

By showing that P(−m) holds true and then deriving P(m) from it Signup and view all the answers

What does it mean when it says 'By induction, P(m) holds for all m ≥ 0'?

All integers greater than or equal to zero satisfy P(m) Signup and view all the answers

What is the role of the division algorithm in proving P(m)?

To show the existence of solutions q, r for m = nq + r Signup and view all the answers

What is the identity element in the group G = Z under the binary operation of addition?

0 Signup and view all the answers

In the group (Z, +), what is the inverse of -5?

5 Signup and view all the answers

Which element does not have an inverse in the set Z under multiplication?

2 Signup and view all the answers

What is the identity element for multiplication in the set of real numbers R?

1 Signup and view all the answers

In k × for k ∈ {Q, R, C}, what is the inverse of 7 under multiplication?

1/7 Signup and view all the answers

Consider G = M2(R), the set of 2×2 matrices with entries in R. What is the identity matrix in this group?

[[1, 0], [0, 1]] Signup and view all the answers

In the group (M2(R), *), what is the inverse of the matrix [[2, 3], [5, 7]]?

[[7, -3], [-5, 2]] Signup and view all the answers

In the group k × for k ∈ {Q, R, C}, which element does not have an inverse under multiplication?

-sqrt(2) Signup and view all the answers

What is the kernel of a group homomorphism according to the provided text?

{g ∈ G | ϕ(g) = eH } Signup and view all the answers

In Lemma 6.31, what condition ensures that a group homomorphism ϕ is injective?

ker ϕ = {eG } Signup and view all the answers

What does ker ϕ denote in terms of the identity element of the group G?

{eG } Signup and view all the answers

Which set is considered the image of a function ϕ: G → H?

{ϕ(g) | g ∈ G} Signup and view all the answers

According to Proposition 6.32, what property holds true about the kernel of a group homomorphism?

It is a subgroup of G. Signup and view all the answers

What condition ensures that ϕ is injective according to Lemma 6.31?

ker ϕ = {eG } Signup and view all the answers

How can we define the image of a function ϕ: G → H?

{ϕ(g) | g ∈ G} Signup and view all the answers

What set does Proposition 6.32 state is a subgroup of H?

{ϕ(g) | g ∈ G} Signup and view all the answers

Which set is defined as the collection of integers?

Z Signup and view all the answers

What does the set $C$ represent?

Complex numbers Signup and view all the answers

In which set would you find the number 0?

Integers Signup and view all the answers

What is the set represented by $N$?

Natural numbers Signup and view all the answers

Which set includes fractions of integers?

Rational numbers Signup and view all the answers

What does the symbol ∅ represent in set theory?

Empty set Signup and view all the answers

Which set includes both irrational and rational numbers?

$R$ Signup and view all the answers

What property distinguishes the natural numbers from other sets mentioned?

They start from 1 Signup and view all the answers

Which number is the greatest common divisor of 114 and 42?

6 Signup and view all the answers

According to Proposition 3.23, if p divides ab, which statement is correct?

p must divide both a and b Signup and view all the answers

Which of the following statements is true according to Lemma 3.22 about gcd(a, p) when p does not divide a?

gcd(a, p) = 1 Signup and view all the answers

What is the definition of a prime number based on Definition 3.21?

A number with only two positive divisors Signup and view all the answers

According to Corollary 3.24 about prime numbers, what can be concluded if a prime p divides a product of integers?

p must divide at least one integer in the product Signup and view all the answers

Based on Theorem 3.25, what can be stated about the fundamental theorem of arithmetic?

Every integer has a unique prime factorization Signup and view all the answers

What does Lemma 3.22 state regarding the relationship between a prime number and a number it divides?

If a prime p does not divide a, then their greatest common divisor is 1 Signup and view all the answers

'Let n be an integer greater than 1.' What type of integer is n based on Definition 3.21?

n can be either prime or composite Signup and view all the answers

What does the proof establish regarding the uniqueness of the identity element in a group?

The identity element is unique Signup and view all the answers

According to the lemma, why is it important that the inverse of an element in a group is unique?

To preserve the uniqueness of the identity element Signup and view all the answers

In a group G, if there are two distinct identity elements e1 and e2, what could be concluded?

Both elements act as identity for all elements in G Signup and view all the answers

Why does the text emphasize the uniqueness of inverses in a group?

To ensure well-defined operations Signup and view all the answers

What does Lemma 5.18 guarantee about the inverses of elements in a group?

Every element has exactly one inverse Signup and view all the answers

In the context of groups, why is it important to prove the uniqueness of the identity element?

To guarantee well-defined operations Signup and view all the answers

How does Lemma 5.18 contribute to ensuring the consistency of group operations?

By proving that inverses are unique Signup and view all the answers

What role does Lemma 5.18 play in verifying the fundamental properties of groups?

Verifying existence and uniqueness of identity and inverses Signup and view all the answers

Which condition is required for a subset H to be a subgroup of G according to Proposition 6.5?

Existence of the identity element Signup and view all the answers

In the context provided, why is N not a subgroup of Z?

It does not contain an identity element. Signup and view all the answers

In Example 6.3, why is H = {e, r, r2} considered a subgroup of G?

Every element has an inverse in H. Signup and view all the answers

What property distinguishes H = {n ∈ Z | n ≥ 0} from being a subgroup of Z?

Having inverses for all elements Signup and view all the answers

Why is N a non-example of a subgroup of Z?

N lacks an identity element under addition. Signup and view all the answers

Why is 1 ∈ H considered not to have an inverse in H under addition?

The additive inverse of 1 is not in H. Signup and view all the answers

What distinguishes N from being considered a subgroup of Z?

Not containing the identity element Signup and view all the answers

Why is H = {n ∈ Z | n ≥ 0} not classified as a subgroup of Z?

There are no inverses within H. Signup and view all the answers

Which operation is defined as multiplication in the set R[x]?

∗ Signup and view all the answers

What is the identity element for addition in the set Z6?

0 Signup and view all the answers

In the set R, C, and Q, what is the identity for multiplication?

1R Signup and view all the answers

Which set forms a ring under addition and multiplication of congruence classes?

Zn Signup and view all the answers

What forms a group under addition with the zero polynomial as the identity element?

R[x] Signup and view all the answers

In which set do we write rs instead of r ∗ s?

{an xn + an−1 xn−1 + · · · + a1 x + a0 | n ≥ 0, ai ∈ R} Signup and view all the answers

What is the inverse of -7 under multiplication in the set of real numbers?

-1 Signup and view all the answers

What are the usual operations defined on Zn to form a ring?

<ul> <li>and *</li> </ul> Signup and view all the answers

What is the formal definition of a group homomorphism?

A function that preserves the binary operations of two groups Signup and view all the answers

In the context provided, what does the division algorithm state?

It expresses an integer as the sum of a quotient and remainder Signup and view all the answers

What form do subgroups of Z take according to Corollary 6.19?

mZ for any integer m Signup and view all the answers

What is the role of propositions and corollaries in mathematical proofs?

To offer extensions or consequences of proven results Signup and view all the answers

How does the concept of closure apply to subgroups?

It states that the product of two elements is always in the subgroup Signup and view all the answers

What distinguishes linear functions in vector spaces from homomorphisms in groups?

Homomorphisms preserve binary operations, while linear functions do not Signup and view all the answers

What does it mean when an element is minimal in a set S according to the text?

The element is not expressible as a combination of other elements in S Signup and view all the answers

In Example 6.21, what set is defined when presented as ⟨m⟩ = mZ?

{m * n} for all integers n Signup and view all the answers

Study Notes

Lecture notes for the course are available from 2020-2021 by Payman Kassaei on the KEATS page, not a substitute for attending lectures.
Additional references for the course include textbooks like "Elementary Number Theory" by David Burton and "Abstract Algebra" by Dummit and Foote.
Sets are defined as collections of elements, examples include integers, natural numbers, rational numbers, real numbers, and complex numbers.
The division algorithm states that for integers m and n, there exist q and r such that m = nq + r where 0 ≤ r < n.
The kernel of a function is a subset of a group where the function maps elements to the identity element of another group.
A group homomorphism ϕ is injective if and only if the kernel of ϕ is {eG}, the identity element of G.
The image of a group homomorphism ϕ is a subgroup of the codomain group, and the kernel of ϕ is a subgroup of the domain group.

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This quiz is based on the lecture notes from 2020-2021 by Payman Kassaei, focusing on Elementary Number Theory. It covers various proofs and examples discussed in the lectures and may reference the book 'Elementary Number Theory' by David Burton.

Elementary Number Theory - Lecture Notes 2020-2021

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Questions and Answers

Which operation forms a group in the set Zn according to the text?

In which set does every element not have an inverse under multiplication?

What property is required for an element in Zn to have a multiplicative inverse?

Which binary operation is defined on Zn by multiplication?

What does the division algorithm state?

What is the condition for an element in Z to have a multiplicative inverse according to the text?

What is the purpose of using induction in proving P(m) for m ≥ 0?

In Z6, which elements do not have multiplicative inverses?

In the proof, what does it mean when it states 'Letting q = q′ + 1 and r = r′ proves P (m)'?

What does the existence of b in Z imply according to Proposition 5.10?

What happens if r′ = 0 in the proof?

$ax \equiv 1$ (mod n) by Proposition 4.25 implies:

How does the proof handle cases when m < 0?

What does it mean when it says 'By induction, P(m) holds for all m ≥ 0'?

What is the role of the division algorithm in proving P(m)?

What is the identity element in the group G = Z under the binary operation of addition?

In the group (Z, +), what is the inverse of -5?

Which element does not have an inverse in the set Z under multiplication?

What is the identity element for multiplication in the set of real numbers R?

In k × for k ∈ {Q, R, C}, what is the inverse of 7 under multiplication?

Consider G = M2(R), the set of 2×2 matrices with entries in R. What is the identity matrix in this group?

In the group (M2(R), *), what is the inverse of the matrix [[2, 3], [5, 7]]?

In the group k × for k ∈ {Q, R, C}, which element does not have an inverse under multiplication?

What is the kernel of a group homomorphism according to the provided text?

In Lemma 6.31, what condition ensures that a group homomorphism ϕ is injective?

What does ker ϕ denote in terms of the identity element of the group G?

Which set is considered the image of a function ϕ: G → H?

According to Proposition 6.32, what property holds true about the kernel of a group homomorphism?

What condition ensures that ϕ is injective according to Lemma 6.31?

How can we define the image of a function ϕ: G → H?

What set does Proposition 6.32 state is a subgroup of H?

Which set is defined as the collection of integers?

What does the set $C$ represent?

In which set would you find the number 0?

What is the set represented by $N$?

Which set includes fractions of integers?

What does the symbol ∅ represent in set theory?

Which set includes both irrational and rational numbers?

What property distinguishes the natural numbers from other sets mentioned?

Which number is the greatest common divisor of 114 and 42?

According to Proposition 3.23, if p divides ab, which statement is correct?

Which of the following statements is true according to Lemma 3.22 about gcd(a, p) when p does not divide a?

What is the definition of a prime number based on Definition 3.21?

According to Corollary 3.24 about prime numbers, what can be concluded if a prime p divides a product of integers?

Based on Theorem 3.25, what can be stated about the fundamental theorem of arithmetic?

What does Lemma 3.22 state regarding the relationship between a prime number and a number it divides?

'Let n be an integer greater than 1.' What type of integer is n based on Definition 3.21?

What does the proof establish regarding the uniqueness of the identity element in a group?

According to the lemma, why is it important that the inverse of an element in a group is unique?

In a group G, if there are two distinct identity elements e1 and e2, what could be concluded?

Why does the text emphasize the uniqueness of inverses in a group?

What does Lemma 5.18 guarantee about the inverses of elements in a group?

In the context of groups, why is it important to prove the uniqueness of the identity element?

How does Lemma 5.18 contribute to ensuring the consistency of group operations?

What role does Lemma 5.18 play in verifying the fundamental properties of groups?

Which condition is required for a subset H to be a subgroup of G according to Proposition 6.5?

In the context provided, why is N not a subgroup of Z?

In Example 6.3, why is H = {e, r, r2} considered a subgroup of G?

What property distinguishes H = {n ∈ Z | n ≥ 0} from being a subgroup of Z?

Why is N a non-example of a subgroup of Z?

Why is 1 ∈ H considered not to have an inverse in H under addition?

What distinguishes N from being considered a subgroup of Z?

Why is H = {n ∈ Z | n ≥ 0} not classified as a subgroup of Z?

Which operation is defined as multiplication in the set R[x]?

What is the identity element for addition in the set Z6?

In the set R, C, and Q, what is the identity for multiplication?

Which set forms a ring under addition and multiplication of congruence classes?

What forms a group under addition with the zero polynomial as the identity element?

In which set do we write rs instead of r ∗ s?

What is the inverse of -7 under multiplication in the set of real numbers?

What are the usual operations defined on Zn to form a ring?

What is the formal definition of a group homomorphism?

In the context provided, what does the division algorithm state?

What form do subgroups of Z take according to Corollary 6.19?

What is the role of propositions and corollaries in mathematical proofs?

How does the concept of closure apply to subgroups?