MATH 201, Algebra II, Exercises (Week 1, due Jan. 11)

(Winter 1999, R. Boltje)

1. Determine Hom(ℤ, ℤ/5ℤ) and Hom(ℤ/5ℤ, ℤ).

2. Show that End(ℚ) ≅ ℚ as rings.

3. Is End(F) ≅ F as rings for every field F?

4. For which n ∈ ℤ is ℤ/nℤ an irreducible ℤ-module?

5. Let R be a ring and let M be an R-module. Show that if M is simple, then End_{R}(M) is a division ring. Is the converse true?

6. Let R be a ring. A left ideal I of a ring R is called a maximal left ideal, if I ≠ R and if R and I are the only left ideals of R containing I. Show that for any R-module M the following are equivalent:

(i) M is an irreducible R-module. (ii) M is isomorphic to R/I for some maximal left ideal I of R.

7. Let R be a ring and let M be an R-module. Show that the following are equivalent:

(i) M is a cyclic R-module. (ii) There exists a left ideal I of R such that M ≅ R/I as R-modules.

(Winter 1999, R. Boltje)

8. Let R be a commutative ring and let M be a finitely generated free Rmodule of R-rank n. Show that if x_{1}, …, x_{n} ∈ M generate M as an R-module, then x_{1}, …, x_{n} is an R-basis of M.

9. An idempotent of a ring is an element e with the property e^{2} = e. Let R be a ring and let M be an R-module. Show that M can be decomposed into an internal direct sum M = U_{1} ⊕ U_{2} of R-submodules U_{1} ≠ 0 ≠ U_{2} if and only if there exists an idempotent in End_{R}(M) which is different from 0 and 1.

10. Are the elements x_{1} = (0, 1, 2), x_{2} = (3, −1, 1), x_{3} = (2, 0, −2) ∈ ℤ^{(3)} (a) a ℚ-basis of ℚ^{(3)}?

(b) a ℤ-basis of ℤ^{(3)}?

11. Let V be the ℝ-vector space of all polynomials

a_{0} + a_{1}λ + … a_{4}λ^{4} ∈ ℝ[λ]

of degree at most 4 and let f: V → V be the ℝ-linear map given by taking the derivative of a polynomial. Find an ℝ-basis X of V such that the matrix representing f with respect to X has the form

A = (\begin{matrix}0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}) .

12. Assume that R is a commutative ring, that M is a free R-module with R-basis X = (x_{1}, …, x_{5}), and that f: M → M is an R-linear map which is represented by the matrix A in Exercise 11 with respect to X. Show that U_{1} := Rx_{1} is an f-stable submodule of M but that there is no f-stable Rsubmodule U_{2} of M such that M = U_{1} ⊕ U_{2}.

13. Let M = ℤ[λ]/(1 + λ + λ^{2})ℤ[λ].

(a) Is M free as ℤ-module?

(b) Is M free as ℤ[λ]-module?

14. Let f: ℝ^{(3)} → ℝ^{(3)} be given by

(\begin{matrix}a \\ b \\ c \end{matrix}) ↦ (\begin{matrix}a − b \\ b − c \\ c − a \end{matrix}) .

Determine the matrix representing f (a) with respect to the canonical ℝ-basis (e_{1}, e_{2}, e_{3}).

(b) with respect to the ℝ-basis (e_{1}, e_{1} + e_{2}, e_{1} + e_{2} + e_{3}).

(Winter 1999, R. Boltje)

15. let R be a domain and let M be an R-module.

(a) Show that tor_{R}(M) is an R-submodule of M.

(b) Show that for each R-module N and each R-linear map f: M → N one has f(tor_{R}(M)) ⊆ tor_{R}(N).

(c) Show that if M is R-free then M is R-torsion free.

16. Let R = ℤ[λ]/(λ^{2} + 1)ℤ[λ] and let M := R. We can consider M as ℤ[λ]-module, as R-module, and as ℤ-module in the usual way. Determine tor_{ℤ[λ]}(M), tor_{R}(M) and tor(M).

17. Give an example of a finitely generated free R-module M over a principal ideal domain R and an R-submodule U of M with rk_{R}(U) = rk_{R}(M) but U ≠ M.

18. Find the invariant factors of the matrix

A = (\begin{matrix}5 & −1 & 4 \\ 3 & −2 & 8 \\ 1 & 1 & −4 \end{matrix}) ∈ M_{3}(ℤ) .

19. Find the invariant factors of the matrix

A = (\begin{matrix}λ^{2} − λ & λ^{3} − 1 & λ^{2} + λ + 1 \\ 2 & λ − 1 & −λ \\ 3λ & λ^{2} − 1 & λ^{2} \end{matrix}) ∈ M_{3}(ℚ[λ]) .

20. Let M = ℤ^{(3)} and let U ⊂ M be the subgroup given by

U = {(x, y, z) ∈ ℤ^{(3)} | x − 2y + 3z = 0} .

Find a ℤ-basis x_{1}, x_{2}, x_{3} of M and numbers d_{1}, d_{2} ∈ ℤ with d_{1} | d_{2} such that d_{1}x_{1}, d_{2}x_{2} is a ℤ-basis of U.

MATH 201, Algebra II, Exercises (Week 4, due Feb. 2)

(Winter 1999, R. Boltje)

21. Let R be a principal ideal domain and let n ∈ ℕ. Show that every matrix A ∈ GL_{n}(R) can be written as a product of matrices of the form T_{ij}(a), D_{i}(u), and P_{ij} as described in Remark 2.4 (c).

22. Let R be a ring, let M be an R-module, and suppose that M = U_{1} ⊕ ⋯⊕U_{n} for R-submodules U_{1}, …, U_{n} of M. Show that, if V_{1} ⊆ U_{1}, …, V_{n} ⊆ U_{n} are further R-submodules and N := V_{1} + ⋯ + V_{n}, then N = V_{1} ⊕ ⋯ ⊕ V_{n} and

M/N ≅ U_{1}/V_{1} ⊕ ⋯ ⊕ U_{n}/V_{n}

as R-modules.

23. Let U ⊆ ℤ^{(n)} be a subgroup generated by elements f_{1}, …, f_{n} with

f_{i} = ∑_{j=1}^{n} a_{ji}e_{i} , a_{ji} ∈ ℤ ,

where e_{1}, …, e_{n} denote the canonical ℤ-basis of ℤ^{(n)}. Assume that d := det(a_{ij}) ≠ 0. Show that |ℤ^{(n)}/U| = |d|.

24. Let M be an R-module over a principal ideal domain R such that dM = 0 for some d ∈ R. Let p ∈ R be a prime element such that M_{p} ≠ 0. Show that then p | d.

25. Let R = ℂ[λ]. How many R-modules M with (λ^{2} − 1)M = 0 and dim_{ℂ} M = 3 do there exist (up to isomorphism)?

26. Determine all abelian groups of order 500 up to isomorphism.

27. Let V be a 3-dimensional ℂ-vector space and let f ∈ End_{ℂ}(V). Assume that with respect to some ordered ℂ-basis (v_{1}, v_{2}, v_{3}) the map f is represented by

A = (\begin{matrix}0 & 1 & 0 \\ −1 & −2 & 0 \\ 3 & 3 & 2 \end{matrix}) .

Determine a Jordan normal form J of the matrix A and an ordered ℂ-basis (w_{1}, w_{2}, w_{3}) of V such that J represents f with respect to (w_{1}, w_{2}, w_{3}).

(Winter 1999, R. Boltje)

28. Let E/F be a field extension and let x ∈ E be transcendental. Show that F [x] ≅ F [X] and F (x) ≅ F (X) as rings.

29. Let F be a field. Show that F contains a smallest subfield P (this means, it is contained in every subfield of F). Show that either P ≅ ℤ/pℤ for some prime p or P ≅ ℚ. Show that in the first case F has characteristic p and in the second case F has characteristic 0.

30. Find the minimal polynomial of x = \sqrt{2}+i over ℚ. Is ℚ(\sqrt{2}i) contained in ℚ(x)? Show that ℚ(x) = ℚ(\sqrt{2}, i). Find the minimal polynomial of x over ℚ(\sqrt{2}i). Try to find as many intermediate fields of ℚ(x)/ℚ as possible.

(Winter 1999, R. Boltje)

31. Let E/F and E′/E be algebraic field extensions. Show that also E′/F is algebraic.

32. Let E/F be a field extension and let S and T be subsets of E.

(a) Show that if S ⊆ T, then F (S) ⊆ F (T).

(b) Show that (F (S))(T) = F (S ∪ T).

(c) Does F (S ∩ T) = F (S) ∩ F (T) hold?

33. Let F be a field, let f(X) ∈ F [X] have degree n > 0, and let E/F be a splitting field of f(X) over F. Show that [E : F] | n!.

34. Find a splitting field of f(X) = X^{4} −2 over ℚ and determine its degree over ℚ.

35. Find all ℚ-algebra homomorphisms from ℚ(\sqrt[4]{2}) to ℚ(\sqrt[4]{2}) and to ℚ(\sqrt[4]{2}, i).

36. Show that each field extension of degree 2 is normal.

37. Let E/F be an algebraic field extension and let K, L be intermediate fields such that K/F and L/F are normal. Show that then also K ∩ L/F and KL/F are normal. (Here, KL is the smallest subfield of E containing K and L. Also KL = K(L) = L(K) = F (K ∪ L).)

(Winter 1999, R. Boltje)

38. Let F be a field of characteristic p > 0. Show that F contains only one p-th root of unity (i.e., an element ζ ∈ F with ζ^{p} = 1), namely 1.

39. Let E/F be a field extension and let f(X), g(X) ∈ F [X]. Let p(X) ∈ F [X] (resp. q(X) ∈ E[X]) be a monic gcd of f(X) and g(X), considered as elements in F [X] (resp. E[X]). Show that p(X) = q(X).

40. Let E/F be an algebraic field extension and let E′ be an intermediate field. Show that E/F is separable if and only if E/E′ and E′/F are separable.

41. Let E/F ba an algebraic field extension. Show that the set

E′ := {a ∈ E | a is separable over F }

is a subfield of E containing F.

42. Let E/F be a finite field extension. Show that the E-vector space Hom_{F} (E, E) as defined in Remark 6.2 has dimension [E : F].

MATH 201, Algebra II, Exercises (Week 8, due March 1)

(Winter 1999, R. Boltje)

43. Let E/F be a finite Galois extension with Galois group G and let E_{1} and E_{2} be intermediate fields corresponding to subgroups H_{1} and H_{2} of G. Show that then H_{1} ∩ H_{2} corresponds to E_{1}E_{2} and < H_{1} ∪ H_{2} > corresponds to E_{1} ∩ E_{2}.

44. Let F ⊆ E ⊆ E_{1} be fields such that E_{1}/F is a finite normal field extension and E/F is a separable field extension. Show that there exists a smallest Galois extension E′/F with E ⊆ E′ ⊆ E_{1}.

45. Let E/F be a finite separable field extension. Show that there exists a Galois extension E′/F such that E ⊆ E′.

46. (a) Show that ℚ(\sqrt[4]{2}, i)/ℚ is a Galois extension.

(b) Compute its Galois group G in terms of the images of \sqrt[4]{2} and i.

(c) Find its action on the roots of f(X) = X^{4} − 2.

(d) Compute all the intermediate fields.

47. Let ζ := e^{2πi/5}. (a)Show that ℚ(ζ + ζ^{−1}) = ℚ(\sqrt{5}).

(b) Find the minimal polynomial of ζ over ℚ.

(c) Show that ℚ(ζ)/ℚ is a Galois extension.

(d) Find all the intermediate fields of ℚ(ζ)/ℚ.

MATH 201, Algebra II, Exercises (Week 9, due March 8)

(Winter 1999, R. Boltje)

48. Let E ⊂ ℂ be the splitting field of f(X) = X^{10} − 5 over ℚ.

(a) Determine [E : ℚ].

(b) Determine the Galois automorphisms of E/ℚ in terms of images of ζ := e^{2πi/5} and \sqrt[10]{5}.

(c) Determine Gal(E/ℚ) as a subgroup of the symmetric group of the roots of f(X).

49. Let F be a field and let f ∈ F [X] ∖ F, such that f and f′ are coprime. Show that f has no multiple roots in any extension field E of F.

50. Compute the cyclotomic polynomial Φ_{12} ∈ ℤ[X].

51. Construct a field with 8 elements and find a primitive 7-th root of unity in it.

52. Let E/F be a field extension and let F′, E_{1}, E_{2} be intermediate fields of E/F such that E_{1} ⊆ E_{2} and such that E_{2}/E_{1} is a Galois extension. Show that then also F′E_{2}/F′E_{1} is a Galois extension and that Gal(F′E_{2}/F′E_{1}) is isomorphic to the subgroup Gal(E_{2}/E_{2} ∩ F′E_{1}) of Gal(E_{2}/E_{1}).