Тест: Abstract Algebra Essentials — 10 въпроса

1. What is the primary focus of abstract algebra as introduced in the course?

A. Applying algebra to physics problems

B. Analyzing algebraic structures like groups, rings, and fields

C. Studying calculus and analysis techniques

D. Solving differential equations

Answer: (B)

Обяснение

Abstract algebra primarily studies algebraic structures such as groups, rings, and fields, focusing on sets with operations satisfying specific axioms, which generalize familiar number systems.

2. What does Cayley's theorem state about finite groups?

A. Every finite group is abelian and can be represented as a matrix group.

B. Any group G can be embedded into a symmetric group Σ(G).

C. Finite groups are necessarily simple groups.

D. Every group G is isomorphic to some ring.

Answer: (B)

Обяснение

Cayley's theorem states that any group G can be embedded into a symmetric group Σ(G), which means G can be thought of as permutations on a set. This is fundamental for understanding group actions and permutation representations.

3. Which property is essential for a subset H of a group G to be considered a normal subgroup?

A. H must be the entire group G

B. H must contain only the identity element

C. H must be a subgroup that is invariant under conjugation by elements of G

D. H must be a subgroup with no elements in common with G

Answer: (C)

Обяснение

A normal subgroup H of G satisfies gHg^{-1} = H for all g in G, meaning it is invariant under conjugation, which is essential for forming quotient groups.

4. Which of the following is TRUE about a prime number?

A. It has exactly three positive divisors.

B. It is only divisible by 1 and itself.

C. It can be divisible by many positive integers less than itself.

D. It must be an even number.

Answer: (B)

Обяснение

A prime number is only divisible by 1 and itself, which distinguishes it from composite numbers. This property is essential for prime factorization and number theory.

5. Under what condition does the residue class ring Z/mZ form a field?

A. When m is a power of 2

B. When m is a prime number

C. When m is an even number

D. When m is a composite number with multiple prime factors

Answer: (B)

Обяснение

Z/mZ is a field if and only if m is prime, because only then do all non-zero elements have multiplicative inverses, making the structure a field.

6. What is a key characteristic of a simple group?

A. It has exactly two elements.

B. It contains no non-trivial normal subgroups.

C. It is always finite.

D. It is abelian and cyclic.

Answer: (B)

Обяснение

A simple group has no non-trivial normal subgroups, making it a fundamental building block in group theory, analogous to prime numbers in number theory.

7. What is necessary for a ring to be classified as a field?

A. It must be non-commutative and contain zero divisors.

B. It must have a multiplicative identity and every non-zero element must have a multiplicative inverse.

C. It must contain only rational numbers.

D. It must be finite and non-associative.

Answer: (B)

Обяснение

A field is a commutative ring with unity in which every non-zero element has an inverse, allowing division to be defined, such as the real or complex numbers.

8. What does the residue class Z/mZ form when m is prime?

A. A ring that is not a field.

B. A field.

C. A non-associative algebra.

D. A group but not a ring.

Answer: (B)

Обяснение

When m is prime, the residue class Z/mZ forms a field because every non-zero element has a multiplicative inverse, a key fact used in modular arithmetic.

9. Who is credited with formulating the theorem stating that any group G can be embedded into a symmetric group?

A. Évariste Galois.

B. Niels Henrik Abel.

C. Arthur Cayley.

D. Emmy Noether.

Answer: (C)

Обяснение

Arthur Cayley is credited with Cayley's theorem, a fundamental result in group theory that demonstrates every group can be represented as a group of permutations.

10. Which of the following correctly describes a Galois extension?

A. A field extension that is algebraic but not necessarily normal.

B. A normal and separable extension with a Galois group of automorphisms fixing the base field.

C. A ring extension with no automorphisms.

D. An extension where elements are algebraic over the complex numbers only.

Answer: (B)

Обяснение

A Galois extension is a field extension that is both normal and separable, and its automorphisms fixing the base field form the Galois group, revealing deep connections between field and group theory.