Fields - fields of fractions of integral domains, characteristics of a field, algebraic extensions, splitting fields, simple extensions - Question Bank

1. The splitting field of an irreducible polynomial p(x) of degree n over a field F is:
A) An extension of degree n over F if p(x) is separable.
B) An extension of degree n over F always.
C) An extension of degree 1 over F if p(x) is irreducible.
D) Not necessarily finite over F.
2. Consider the extension Q(∛2, ω) over Q, where ω is a primitive cube root of unity. What is the degree of this extension?
A) 6
B) 3
C) 2
D) 9
3. Let K be an extension field of F. If [K:F] is finite, then K is:
A) An algebraic extension of F.
B) A transcendental extension of F.
C) Equal to F.
D) Not necessarily algebraic over F.
4. If a field has characteristic p > 0, what can be said about the polynomial x^p - x?
A) Its roots form a finite field.
B) It has no roots in the field.
C) It is irreducible.
D) Its degree is p-1.
5. Let F be a field. The field of fractions of F[x] is isomorphic to:
A) The field of rational functions F(x).
B) The field F[x] itself.
C) The algebraic closure of F.
D) The field F.
6. What is the field of fractions of the integral domain Z[√2] = {a + b√2 | a, b ∈ Z}?
A) Q(√2)
B) R
C) Z[√2]
D) C
7. If K is the splitting field of a separable polynomial p(x) over F, then [K:F] is equal to:
A) The degree of p(x).
B) The number of distinct roots of p(x).
C) The number of irreducible factors of p(x).
D) The degree of the minimal polynomial of a root of p(x).
8. Let α be an element of an extension field K of F. If α is algebraic over F, then F(α) is:
A) A finite extension of F.
B) An infinite extension of F.
C) A transcendental extension of F.
D) Equal to F.
9. The set of all algebraic numbers over Q forms:
A) A field, but not algebraically closed.
B) An algebraically closed field.
C) A ring, but not a field.
D) A field that is not algebraic over Q.
10. If F is a field, what can be said about the characteristic of any subfield of F?
A) It is the same as the characteristic of F.
B) It can be different from the characteristic of F.
C) It must be 0.
D) It must be a prime number.
11. Let K be the splitting field of x^2 + 1 over R. What is K?
A) The complex numbers C.
B) The real numbers R.
C) R(i).
D) Q(i).
12. What is the characteristic of the field Z_4 (integers modulo 4)?
A) It is not a field.
B) 4
C) 2
D) 0
13. Let F be a field of characteristic 0. If α is algebraic over F, then the minimal polynomial of α over F is:
A) Unique and irreducible.
B) Not necessarily unique.
C) Always linear.
D) Always of degree 0.
14. What is the field of fractions of the ring of integers Z?
A) The rational numbers Q.
B) The real numbers R.
C) The integers Z.
D) The complex numbers C.
15. If α is transcendental over F, what is the degree of the extension F(α) over F?
A) Infinite
B) 1
C) 2
D) Degree of the minimal polynomial of α
16. Consider the extension Q(√2, √3) over Q. What is [Q(√2, √3):Q]?
A) 4
B) 2
C) 6
D) 3
17. Let F = Z_p(t) be the field of rational functions in t over Z_p. What is the characteristic of F?
A) p
B) 0
C) 1
D) Infinite
18. If K is an algebraic extension of F, and L is an algebraic extension of K, then L is algebraic over F. This is known as the:
A) Tower law for algebraic extensions.
B) Transitivity of algebraic extensions.
C) Multiplicativity of degrees for algebraic extensions.
D) Existence of splitting fields.
19. What is the smallest field containing F and all the roots of a polynomial p(x) ∈ F[x]?
A) The splitting field of p(x) over F.
B) The algebraic closure of F.
C) The field F(α) where α is any root of p(x).
D) The field F itself.
20. Let α be algebraic over F. The degree of F(α) over F is equal to the degree of the minimal polynomial of α over F. This statement is:
A) True.
B) False, it is always greater than the degree of the minimal polynomial.
C) False, it is always less than the degree of the minimal polynomial.
D) False, unless F(α) = F.
21. If F is a field, what is its characteristic?
A) Either 0 or a prime number.
B) Always 0.
C) Always a positive integer.
D) Always 1.
22. Which of the following is NOT an integral domain?
A) The set of integers Z.
B) The set of polynomials R[x].
C) The ring of 2x2 matrices with real entries.
D) The set of Gaussian integers Z[i].
23. Let K be the splitting field of x^3 - 2 over Q. What is [K:Q]?
A) 6
B) 3
C) 9
D) 2
24. What is the splitting field of the polynomial x^p - a over F, where F has characteristic p > 0?
A) A purely inseparable extension.
B) A purely separable extension.
C) The field F itself.
D) The algebraic closure of F.
25. Let K be a finite extension of F, and let L be a finite extension of K. Then:
A) L is a finite extension of F, and [L:F] = [L:K][K:F].
B) L is an algebraic extension of F, and [L:F] = [L:K] + [K:F].
C) L is a finite extension of F, and [L:F] = [L:K] / [K:F].
D) L is an algebraic extension of F, and [L:F] = [L:K] - [K:F].
26. Consider the field extension Q(i) over Q, where i^2 = -1. What is the degree [Q(i):Q]?
A) 2
B) 1
C) i
D) 4
27. Let K be an extension of F. If α ∈ K is algebraic over F, then the degree of the minimal polynomial of α over F is equal to:
A) [F(α):F]
B) [K:F]
C) The number of conjugates of α.
D) The number of roots of the minimal polynomial in K.
28. If a field F has characteristic p > 0, then for any elements a, b in F, which property holds?
A) (a + b)^p = a^p + b^p
B) (a + b)^p = a^p + b^p + p(ab)^(p-1)
C) (a + b)^p = a^p + p a^(p-1)b + ... + b^p
D) (a + b)^p = a^p + b^p + p ab
29. What is the characteristic of a field F if 1 + 1 = 0 in F?
A) 2
B) 0
C) 1
D) Undefined
30. The field of fractions of an integral domain R is constructed by:
A) Forming equivalence classes of fractions a/b where a, b are in R and b ≠ 0.
B) Taking all elements of R and adding their inverses.
C) Adjoining all roots of all polynomials in R.
D) Considering all possible finite products of elements in R.
31. Let F be a field. What is the field of fractions of the integral domain F[x]?
A) The field of rational functions in x over F, denoted F(x).
B) The field of polynomials in x over F, F[x].
C) The algebraic closure of F.
D) The field F itself.
32. Consider the extension Q(∛2) over Q. Is this a finite extension?
A) Yes, because the minimal polynomial of ∛2 over Q is x^3 - 2, which has degree 3.
B) No, because ∛2 is not rational.
C) No, because Q(∛2) is an infinite field.
D) Yes, because Q(∛2) is a subfield of R.
33. What is the relationship between algebraic extensions and finite extensions?
A) Every finite extension is algebraic, but the converse is not always true.
B) Every algebraic extension is finite, but the converse is not always true.
C) Finite and algebraic extensions are equivalent concepts.
D) There is no direct relationship between finite and algebraic extensions.
34. If K is a splitting field of a polynomial p(x) over F, and L is an intermediate field such that F ⊆ L ⊆ K, is L necessarily the splitting field of p(x) over F?
A) No, not necessarily.
B) Yes, always.
C) Yes, if p(x) is irreducible.
D) Yes, if L = F(roots of p(x)).
35. Let K be the splitting field of x^2 - 2 over Q. What is K?
A) Q(√2)
B) Q
C) R
D) Q(√2, √3)
36. For any non-constant polynomial p(x) in F[x], does a splitting field always exist?
A) Yes, it always exists.
B) No, it only exists if F is algebraically closed.
C) No, it only exists if p(x) is irreducible.
D) No, it only exists if F has characteristic zero.
37. What is a splitting field of a polynomial p(x) over a field F?
A) The smallest extension field K of F such that p(x) splits into linear factors in K[x].
B) Any extension field K of F such that p(x) splits into linear factors in K[x].
C) The field F itself if p(x) splits in F[x].
D) The field of fractions of F[x] / <p(x)>.
38. Consider the extension Q(√2) over Q. What is the degree of this extension?
A) 2
B) 1
C) √2
D) 4
39. If [K:F] = 1, what is the relationship between K and F?
A) K = F
B) K is a proper extension of F
C) F is an algebraic extension of K
D) K is a transcendental extension of F
40. The degree of a field extension K over F, denoted [K:F], is defined as:
A) The dimension of K as a vector space over F.
B) The number of elements in K divided by the number of elements in F.
C) The degree of the minimal polynomial of any element in K over F.
D) The number of intermediate fields between F and K.
41. What is the minimal polynomial of an element α algebraic over a field F?
A) The monic polynomial of least degree in F[x] that has α as a root.
B) Any polynomial in F[x] that has α as a root.
C) The polynomial of highest degree in F[x] that has α as a root.
D) The characteristic polynomial of α.
42. If every element of an extension field K of F is algebraic over F, then K is called:
A) An algebraic extension of F
B) A transcendental extension of F
C) A simple extension of F
D) A finite extension of F
43. An element α in an extension field K of F is said to be algebraic over F if:
A) There exists a non-zero polynomial p(x) with coefficients in F such that p(α) = 0.
B) α is transcendental over F.
C) The minimal polynomial of α over F has degree 1.
D) K is a finite extension of F.
44. Let K be an extension of F. If K = F(α) for some element α in K, what is K called?
A) A simple extension of F
B) A finite extension of F
C) An algebraic extension of F
D) A splitting field extension of F
45. What is a field extension?
A) A field K such that F is a subfield of K.
B) A field K such that F is a subring of K.
C) A field K such that F is an ideal of K.
D) A field K such that F is isomorphic to K.
46. If F is a field of characteristic p > 0, what is true about the additive subgroup generated by the multiplicative identity 1?
A) It is isomorphic to Zp.
B) It is isomorphic to Z.
C) It is the trivial subgroup {0}.
D) It is isomorphic to Q.
47. What is the characteristic of the field of rational numbers (Q)?
A) 1
B) 0
C) Infinity
D) Undefined
48. What is the characteristic of the field Zp, where p is a prime number?
A) 1
B) 0
C) p
D) p-1
49. Which of the following is a field of fractions of the integral domain of integers (Z)?
A) The set of rational numbers (Q)
B) The set of real numbers (R)
C) The set of integers modulo 5 (Z5)
D) The set of polynomials with integer coefficients (Z[x])
50. What is the defining property of an integral domain?
A) It contains a multiplicative identity and every non-zero element has a multiplicative inverse.
B) It is a commutative ring with unity and no zero divisors.
C) It is a field.
D) It has a finite number of elements.
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