Multiple Choice Questions
Abstract Algebra: Fields and Galois Theory
Topic: Fields and Galois Theory
Grade: 12
Question 1:
Which of the following is a field?
a) ℤ
b) ℝ
c) ℚ[x]
d) ℤ[i]
Answer: b) ℝ
Explanation: A field is a set with two binary operations, addition and multiplication, that satisfy certain axioms. The set of real numbers ℝ is a field because addition and multiplication of real numbers are commutative, associative, have identities (0 and 1, respectively), and every non-zero real number has a multiplicative inverse. For example, the real numbers can be added and multiplied just like ordinary numbers.
Question 2:
Let F be a field and let a ∈ F. If a² = a, then a is called a:
a) zero divisor
b) unit
c) idempotent
d) integral element
Answer: c) idempotent
Explanation: An element a ∈ F is called idempotent if a² = a. In other words, the element is unchanged when squared. For example, in the field of real numbers, 0 and 1 are idempotent elements because 0² = 0 and 1² = 1.
Question 3:
Which of the following is a characteristic of a field?
a) Every element has an additive inverse
b) Every non-zero element has a multiplicative inverse
c) Addition is associative
d) Multiplication is commutative
Answer: b) Every non-zero element has a multiplicative inverse
Explanation: One of the fundamental properties of a field is that every non-zero element has a multiplicative inverse. This means that for every non-zero element a, there exists an element b such that a * b = 1. For example, in the field of rational numbers ℚ, the multiplicative inverse of 3 is 1/3, because 3 * (1/3) = 1.
Question 4:
Let F be a field and let a, b ∈ F. If a * b = b * a for all a, b ∈ F, then F is called a:
a) commutative field
b) finite field
c) Galois field
d) perfect field
Answer: a) commutative field
Explanation: A field is called commutative if the multiplication operation is commutative, meaning that a * b = b * a for all elements a and b in the field. For example, the set of real numbers ℝ is a commutative field because multiplication of real numbers is commutative.
Question 5:
Which of the following is a finite field?
a) ℤ
b) ℝ
c) ℚ
d) ℤ₅
Answer: d) ℤ₅
Explanation: A finite field is a field with a finite number of elements. The set of integers ℤ, real numbers ℝ, and rational numbers ℚ are all infinite fields. However, the set ℤ₅, also known as the integers modulo 5, is a finite field because it consists of the elements {0, 1, 2, 3, 4}. Addition and multiplication in ℤ₅ are performed modulo 5, meaning that any result greater than 4 is reduced by subtracting 5. For example, in ℤ₅, 3 + 4 = 2, and 3 * 4 = 2.
Question 6:
Which of the following is true for a Galois field?
a) It is always a finite field
b) It is always an infinite field
c) It is always a commutative field
d) It is always a perfect field
Answer: a) It is always a finite field
Explanation: A Galois field, also known as a finite field, is a field with a finite number of elements. Every Galois field is finite, but not every finite field is a Galois field. For example, ℤ₅ is a Galois field because it has a finite number of elements.
Question 7:
Let F be a field and let p(x) be a polynomial of degree n with coefficients in F. If p(x) has n distinct roots in an extension field of F, then F is called:
a) algebraically closed
b) perfect
c) separable
d) finite
Answer: a) algebraically closed
Explanation: A field F is called algebraically closed if every non-constant polynomial with coefficients in F has a root in F. In this case, if the polynomial p(x) has n distinct roots in an extension field of F, it means that every root of p(x) is also in F, making F algebraically closed. For example, the field of complex numbers ℂ is algebraically closed because every non-constant polynomial with complex coefficients has a root in ℂ.
Question 8:
Which of the following is a separable polynomial?
a) x² – 2x + 1
b) x³ – x² + x – 1
c) x⁴ + 2x² + 1
d) x⁵ + x
Answer: a) x² – 2x + 1
Explanation: A polynomial is called separable if all of its irreducible factors are distinct. In other words, the polynomial has no repeated roots. The polynomial x² – 2x + 1 has distinct roots 1 and 1, making it separable. For example, x² – 2x + 1 factors into (x – 1)(x – 1), where both factors are distinct irreducible polynomials.
Question 9:
Let F be a field and let p(x) be a polynomial of degree n with coefficients in F. If p(x) has no repeated roots in an extension field of F, then F is called:
a) algebraically closed
b) perfect
c) separable
d) finite
Answer: c) separable
Explanation: A field F is called separable if every irreducible polynomial over F has no repeated roots in any extension field of F. In this case, if the polynomial p(x) has no repeated roots in an extension field of F, it means that every irreducible factor of p(x) is distinct, making F separable. For example, the field of rational numbers ℚ is separable because every irreducible polynomial over ℚ has no repeated roots.
Question 10:
Which of the following is a perfect field?
a) ℤ
b) ℝ
c) ℚ
d) ℤ₅
Answer: d) ℤ₅
Explanation: A perfect field is a field in which every irreducible polynomial has no repeated roots in any extension field. The field ℤ₅ is a perfect field because every irreducible polynomial over ℤ₅ has no repeated roots.
Question 11:
Let F be a field and let α be an element in an extension field of F. If α is a root of an irreducible polynomial over F, then F(α) is called a:
a) splitting field
b) normal extension
c) Galois extension
d) separable extension
Answer: a) splitting field
Explanation: A splitting field is an extension field of a given field F in which every polynomial over F splits into linear factors. In this case, since α is a root of an irreducible polynomial over F, the field F(α) is called a splitting field. For example, if F is the field of rational numbers ℚ and α is a root of the irreducible polynomial x² – 2, then ℚ(α) is the splitting field of x² – 2 over ℚ.
Question 12:
Which of the following is a normal extension?
a) ℚ(√2)
b) ℚ(√3)
c) ℚ(√2, √3)
d) ℚ(√2 + √3)
Answer: c) ℚ(√2, √3)
Explanation: An extension field F(α) of a given field F is called normal if every irreducible polynomial over F with a root in F(α) splits completely over F(α). In this case, ℚ(√2, √3) is a normal extension because every irreducible polynomial over ℚ with a root in ℚ(√2, √3) splits completely over ℚ(√2, √3). For example, the irreducible polynomial x² – 2 over ℚ has a root √2 in ℚ(√2, √3), and it splits into linear factors (x – √2)(x + √2) over ℚ(√2, √3).
Question 13:
Let F be a field and let α be an element in an extension field of F. If α is separable over F, then α is a root of a:
a) reducible polynomial
b) irreducible polynomial
c) polynomial with repeated roots
d) polynomial with no repeated roots
Answer: b) irreducible polynomial
Explanation: An element α in an extension field F(α) is separable over F if the minimal polynomial of α over F has no repeated roots. In this case, α is a root of an irreducible polynomial over F. For example, if α is a root of the irreducible polynomial p(x), then p(x) is the minimal polynomial of α, and α is separable over F.
Question 14:
Which of the following is a Galois extension?
a) ℚ(√2)
b) ℚ(√3)
c) ℚ(√2, √3)
d) ℚ(√2 + √3)
Answer: c) ℚ(√2, √3)
Explanation: An extension field F(α) of a given field F is called a Galois extension if it is a normal and separable extension. In this case, ℚ(√2, √3) is a Galois extension because it is both normal and separable. For example, every irreducible polynomial over ℚ with a root in ℚ(√2, √3) splits completely over ℚ(√2, √3), and every element in ℚ(√2, √3) is separable over ℚ.
Question 15:
Let F be a field and let α, β be elements in an extension field of F. If α and β are algebraic over F, then α + β is also:
a) algebraic over F
b) transcendental over F
c) separable over F
d) idempotent over F
Answer: a) algebraic over F
Explanation: If α and β are algebraic over a field F, it means that there exist non-zero polynomials with coefficients in F such that these polynomials have α and β as roots, respectively. In this case, α + β is also algebraic over F because there exists a polynomial with coefficients in F whose root is α + β. For example, if α is a root of the polynomial p(x) and β is a root of the polynomial q(x), then α + β is a root of the polynomial p(x) + q(x).