TRB PG Assistant - Galois theory - basic elements, finite fields One Line Questions & Answers

1. The number of distinct irreducible polynomials of degree d over GF(q) is given by: — (1/d) * sum_{k|d} mu(d/k) * q^k

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

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

4. The field Q(sqrt(2), sqrt(3)) over Q has degree: — 4

5. Let K be the splitting field of x^4 - 2 over Q. The degree [K:Q] is: — 8

6. 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 cyclic group of order n, where q = p^n.

7. What is the multiplicative group of a finite field GF(q)? — A cyclic group of order q-1.

8. What is a field extension? — A field K such that F is a subfield of K.

9. What is a finite field? — A field with a finite number of elements.

10. How many elements does a finite field always have? — A power of a prime number (p^n).

11. What is an algebraic element over a field F? — An element that is a root of some non-zero polynomial with coefficients in F.

12. The Galois group of Q(i) over Q is isomorphic to: — C_2 (cyclic group of order 2)

13. A field is called algebraically closed if: — Every polynomial with coefficients in the field has a root in the field.

14. If p is a prime, the polynomial x^p - x over GF(p) has: — Exactly p distinct roots, which are all the elements of GF(p).

15. For a finite field GF(q), the Frobenius automorphism is an element of which group? — Gal(GF(q)/GF(p))

16. Which of the following is NOT a finite field? — GF(15)

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

18. The field with q elements, where q is a prime power, is denoted by: — GF(q) or F_q

19. Let F = GF(2). The polynomial x^3 + x + 1 is: — Irreducible over F.

20. Let F = GF(2). The polynomial x^4 + x + 1 is: — Irreducible over F.

21. A polynomial is called separable if: — It has distinct roots in its splitting field.

22. A finite extension K/F is called a Galois extension if: — K is the splitting field of a separable polynomial over F, and the extension is normal and separable.

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

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

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

26. The order of the multiplicative group of GF(p^n) is: — p^n - 1

27. What is the degree of the extension Q(zeta_p) over Q, where p is a prime? — p-1

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

29. What is the splitting field of x^3 - 2 over Q? — Q(cbrt(2), omega), where omega is a primitive cube root of unity.

30. The splitting field of x^4 - 1 over Q is: — Q(i)

31. All finite fields have the property that all polynomials over them are: — Separable

32. The fundamental theorem of Galois theory establishes a correspondence between: — Subfields of the splitting field containing the base field and subgroups of the Galois group.

33. The constructibility of a regular n-gon using ruler and compass is related to Galois theory via: — The Galois group of the cyclotomic field Q(zeta_n) over Q.

34. What is the Galois group of a polynomial over a field F? — The group of all automorphisms of the splitting field of the polynomial that fix F.

35. 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: — The group of units modulo n, (Z/nZ)*.

36. What is the Frobenius automorphism for a finite field GF(p^n)? — The map x -> x^p.

37. What is the minimal polynomial of an element 'a' over a field F? — The monic irreducible polynomial of smallest degree with coefficients in F that has 'a' as a root.

38. The degree of a field extension K over F, denoted [K:F], represents: — The dimension of K as a vector space over F.

39. The Galois group of a polynomial is trivial (consists only of the identity element) if and only if: — The polynomial splits into linear factors over the base field.

40. The field GF(p) is isomorphic to: — The ring of integers modulo p, Z_p.

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

42. In the context of Galois theory, what is the primary object of study when considering a polynomial over a field? — The roots of the polynomial and their permutations.

43. What is the characteristic of a field F? — The smallest positive integer n such that n * 1 = 0, where 1 is the multiplicative identity, or 0 if no such positive integer exists.

44. What is the prime subfield of any field F? — The field of rational numbers if the characteristic is 0, or the field with p elements if the characteristic is p.

45. The Galois group of the splitting field of x^3 - 2 over Q is isomorphic to: — The symmetric group S_3.

46. Which of the following is a consequence of the Abel-Ruffini theorem, proven using Galois theory? — There is no general algebraic solution (using radicals) to polynomial equations of degree five or higher.

47. 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: — Tower Law

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

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

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

Galois theory - basic elements, finite fields - One Line Questions