TRB PG Assistant - Galois theory - basic elements, finite fields Online Test (MCQ Quiz)

1. In the context of Galois theory, what is the primary object of study when considering a polynomial over a field?

A) The roots of the polynomial and their permutations. B) The coefficients of the polynomial. C) The degree of the polynomial. D) The number of irreducible factors of the polynomial.

2. What is a field extension?

A) A field K such that F is a subfield of K. B) A ring R such that F is a subfield of R. C) A group G such that F is a subgroup of G. D) A vector space V such that F is a subspace of V.

3. The degree of a field extension K over F, denoted [K:F], represents:

A) The number of elements in K divided by the number of elements in F. B) The dimension of K as a vector space over F. C) The number of automorphisms of K that fix F. D) The number of prime ideals in the ring K[x] modulo F[x].

4. What is the Galois group of a polynomial over a field F?

A) The group of all automorphisms of the splitting field of the polynomial that fix F. B) The group of all automorphisms of the polynomial itself. C) The group of permutations of the roots of the polynomial. D) The group of units in the splitting field.

5. A field is called algebraically closed if:

A) Every polynomial with coefficients in the field has a root in the field. B) The field contains all roots of unity. C) The field has a finite number of elements. D) The field is a subfield of the real numbers.

6. What is a finite field?

A) A field with a finite number of elements. B) A field whose characteristic is finite. C) A field that is a subfield of the real numbers. D) A field that is algebraically closed.

7. What is the characteristic of a field F?

A) The smallest positive integer n such that n * 1 = 0, where 1 is the multiplicative identity, or 0 if no such positive integer exists. B) The number of elements in the prime subfield of F. C) The degree of the minimal polynomial of any element in F. D) The number of distinct roots of the polynomial x^p - x for some prime p.

8. What is the prime subfield of any field F?

A) The smallest subfield of F. B) The intersection of all subfields of F. C) The field generated by the element 1. D) The field of rational numbers if the characteristic is 0, or the field with p elements if the characteristic is p.

9. The field with q elements, where q is a prime power, is denoted by:

A) GF(q) or F_q B) Z_q C) Q(q) D) R_q

10. How many elements does a finite field always have?

A) A power of a prime number (p^n). B) A prime number. C) An even number. D) An odd number.

11. What is the characteristic of the finite field GF(p^n) where p is a prime and n >= 1?

A) p B) n C) p^n D) 1

12. The field GF(p) is isomorphic to:

A) The ring of integers modulo p, Z_p. B) The field of rational numbers, Q. C) The field of real numbers, R. D) The field of complex numbers, C.

13. For any prime p and positive integer n, there exists a unique (up to isomorphism) finite field with p^n elements. This statement is:

A) True B) False C) True only if n=1 D) True only if p is a Mersenne prime

14. What is the multiplicative group of a finite field GF(q)?

A) A cyclic group of order q-1. B) A cyclic group of order q. C) An abelian group of order q-1. D) A non-abelian group of order q.

15. Consider the polynomial x^2 + 1 over the field GF(2). What are its roots?

A) 0 and 1 B) 1 and 1 C) 0 and 0 D) No roots in GF(2)

16. Let F = GF(3). Consider the polynomial x^2 + 1. Which of the following is an irreducible polynomial over F?

A) x^2 + 1 B) x - 1 C) x^2 - 1 D) x^3 - 1

17. If p is a prime, the polynomial x^p - x over GF(p) has:

A) Exactly p distinct roots, which are all the elements of GF(p). B) Exactly p-1 distinct roots. C) Exactly one root. D) No roots.

18. The splitting field of an irreducible polynomial of degree n over a finite field GF(p) is isomorphic to:

A) GF(p^n) B) GF(p) C) GF(n) D) GF(p^p)

19. Let K be a finite field with q elements. The Galois group Gal(K/GF(p)), where p is the characteristic of K, is isomorphic to:

A) A cyclic group of order n, where q = p^n. B) A cyclic group of order p. C) The trivial group. D) The symmetric group S_n.

20. What is the Frobenius automorphism for a finite field GF(p^n)?

A) The map x -> x^p. B) The map x -> x^(p^n). C) The map x -> x^(p-1). D) The map x -> x + p.

21. The Frobenius automorphism for GF(p^n) generates the Galois group Gal(GF(p^n)/GF(p)). This statement is:

A) True B) False C) True only if n is prime D) False, the generator is x -> x^(p^k) for some k

22. Consider the field extension Q(sqrt(2)) over Q. The degree of this extension is:

A) 2 B) 1 C) 4 D) Infinitely many

23. What is the splitting field of x^3 - 2 over Q?

A) Q(cbrt(2), omega), where omega is a primitive cube root of unity. B) Q(cbrt(2)) C) Q(sqrt(2)) D) Q(omega)

24. The Galois group of the splitting field of x^3 - 2 over Q is isomorphic to:

A) The symmetric group S_3. B) The cyclic group C_3. C) The cyclic group C_2. D) The trivial group.

25. A polynomial is called separable if:

A) It has distinct roots in its splitting field. B) It has only one root. C) Its degree is prime. D) It is irreducible.

26. All finite fields have the property that all polynomials over them are:

A) Separable B) Inseparable C) Irreducible D) Reducible

27. What is the minimal polynomial of an element 'a' over a field F?

A) The monic irreducible polynomial of smallest degree with coefficients in F that has 'a' as a root. B) Any polynomial with coefficients in F that has 'a' as a root. C) The polynomial of largest degree with coefficients in F that has 'a' as a root. D) The characteristic polynomial of the matrix associated with 'a'.

28. If K is a finite extension of a field F, and L is a finite extension of K, then [L:F] = [L:K] * [K:F]. This is the theorem of:

A) Tower Law B) Lagrange C) Galois D) Cauchy

29. The Galois group of a polynomial is trivial (consists only of the identity element) if and only if:

A) The polynomial splits into linear factors over the base field. B) The polynomial is irreducible. C) The polynomial has degree 2. D) The polynomial has exactly one root.

30. Which of the following is NOT a finite field?

A) GF(7) B) GF(16) C) GF(15) D) GF(25)

31. What is the characteristic of GF(p^n)?

A) p B) n C) p^n D) 1

32. The number of distinct irreducible polynomials of degree d over GF(q) is given by:

A) (1/d) * sum_{k|d} mu(d/k) * q^k B) q^d C) d^q D) q

33. Let F = GF(2). The polynomial x^3 + x + 1 is:

A) Irreducible over F. B) Reducible over F. C) A product of two linear factors. D) Equal to x^3 + x + 1 + 1.

34. The splitting field of x^4 - 1 over Q is:

A) Q(i) B) Q(sqrt(2)) C) Q(cbrt(1)) D) Q

35. The Galois group of Q(i) over Q is isomorphic to:

A) C_2 (cyclic group of order 2) B) C_1 (trivial group) C) S_2 D) C_4

36. What is an algebraic element over a field F?

A) An element that is a root of some non-zero polynomial with coefficients in F. B) An element that is not transcendental over F. C) An element that belongs to an algebraically closed field. D) An element that belongs to a finite field.

37. Let K be the splitting field of a polynomial P(x) over F. The Galois group Gal(K/F) acts on:

A) The roots of P(x). B) The coefficients of P(x). C) The field K itself. D) The set of irreducible factors of P(x).

38. A finite field GF(q) is a vector space of dimension n over its prime subfield GF(p) if:

A) q = p^n B) q = n C) q = p D) n = p

39. The order of the multiplicative group of GF(p^n) is:

A) p^n - 1 B) p^n C) p - 1 D) n

40. Let F = GF(2). The polynomial x^4 + x + 1 is:

A) Irreducible over F. B) Reducible over F. C) A product of three linear factors. D) A square of a polynomial.

41. The Galois group of the cyclotomic field Q(zeta_n) over Q, where zeta_n is a primitive n-th root of unity, is isomorphic to:

A) The group of units modulo n, (Z/nZ)*. B) The cyclic group C_n. C) The symmetric group S_n. D) The additive group of Z/nZ.

42. What is the degree of the extension Q(zeta_p) over Q, where p is a prime?

A) p-1 B) p C) 1 D) p+1

43. The fundamental theorem of Galois theory establishes a correspondence between:

A) Subfields of the splitting field containing the base field and subgroups of the Galois group. B) Roots of the polynomial and elements of the Galois group. C) Irreducible factors and normal subgroups. D) Field extensions and vector spaces.

44. A finite extension K/F is called a Galois extension if:

A) K is the splitting field of a separable polynomial over F, and the extension is normal and separable. B) K is algebraically closed. C) F is algebraically closed. D) K has finitely many elements.

45. For a finite field GF(q), the Frobenius automorphism is an element of which group?

A) Gal(GF(q)/GF(p)) B) Aut(GF(q)) C) GL_n(GF(p)) D) SL_n(GF(p))

46. What is the degree of the field extension GF(p^m) over GF(p^n), where n divides m?

A) m/n B) m*n C) m-n D) m+n

47. The constructibility of a regular n-gon using ruler and compass is related to Galois theory via:

A) The Galois group of the cyclotomic field Q(zeta_n) over Q. B) The degree of the field extension Q(sqrt(2)) over Q. C) The characteristic of finite fields. D) The number of elements in the multiplicative group of a finite field.

48. Which of the following is a consequence of the Abel-Ruffini theorem, proven using Galois theory?

A) There is no general algebraic solution (using radicals) to polynomial equations of degree five or higher. B) All polynomial equations of degree three have solutions in radicals. C) Quadratic equations can always be solved using integers. D) Cubic equations can be solved using only addition and multiplication.

49. The field Q(sqrt(2), sqrt(3)) over Q has degree:

A) 4 B) 2 C) 3 D) 6

50. Let K be the splitting field of x^4 - 2 over Q. The degree [K:Q] is:

A) 8 B) 4 C) 6 D) 2

Galois theory - basic elements, finite fields - Online Test

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