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==HOMEWORK 9 (due Nov 20)== | ==HOMEWORK 9 (due Nov 20)== | ||

0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.] | |||

1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring. | 1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring. | ||

1a. Show that | 1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.) | ||

1b. Show that | 1b. Show that every Boolean ring is commutative. | ||

## Revision as of 01:27, 26 November 2012

**Math 741**

Algebra

Prof: Jordan Ellenberg

Grader: Evan Dummit

Ellenberg's office hours: Friday 3pm.

Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.

This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.

## SYLLABUS

In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course.

**WEEK 1**:

Definition of group. Associativity. Inverse.

Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators.

Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.

**WEEK 2**:

The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order.

Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.

The sign homomorphism S_n -> +-1.

**WEEK 3**

Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.

Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.

Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.

**WEEK 4**

More on groups with presentations.

Second and third isomorphism theorems.

Semidirect products.

**WEEK 5**

Group actions, orbits, and stabilizers.

Orbit-stabilizer theorem.

Cayley's theorem.

Cauchy's theorem.

**WEEK 6**

Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)

Classification of finite abelian groups and finitely generated abelian groups.

Composition series and the Jordan-Holder theorem (which we state but don't prove.)

The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)

**WEEK 7**

Simplicity of A_n.

Nilpotent groups (main example: the Heisenberg group)

Derived series and lower central series.

Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.

**WEEK 8**

Introduction to representation theory.

**WEEK 10**

Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.

Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.

Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's *Algebra*.

## HOMEWORK 1 (due Sep 20)

1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.

2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.

3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.

a) For each n, prove that Q/Z has a subgroup of order n.

b) Prove that Q/Z is a *divisible* group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)

c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)

4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.

5. Let H be a subgroup of G, and let N_G(H), the *normalizer* of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.

6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T.

7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).

8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a *double coset* of the pair (H,H').

a) Show that G decomposes as a disjoint union of double cosets of (H,H').

b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).

c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).

## HOMEWORK 2 (due Sep 27)

1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.

2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.

3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)

4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.

5. The **quaternion group** Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.

6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z.

7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.

## HOMEWORK 3 (due Oct 4)

1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a *dihedral group.* and is denoted D_n (or sometimes D_{2n}).

1a. Compute the center of D_n. (Note that the answer depends on n!) 1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.

2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.

3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.

4. The *affine linear group* of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.

5. The *ordinary triangle group* T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.

6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n. 6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?

(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)

7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).

8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.

9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.

9a. Show that G' is a normal subgroup of G.
9b. Show that G/G' is an abelian group.
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is *perfect*.

10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?

## HOMEWORK 4 (due Oct 16)

1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.

1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)

1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the *normal core* of H_x.)

2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.

2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.

2b. A *unipotent matrix* is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.

2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.

2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.

2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?

3. Let H be a subgroup of G of index 2. Prove that H is normal.

4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.

5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.

6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.

7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)

8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.

8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.

8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.

(For those reading MacLane, this constitutes a proof that abelianization is a **functor** from the category of groups to the category of abelian groups.)

## HOMEWORK 5 (due Oct 23)

1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)

2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X.

2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.

2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.

3. A *central extension* of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.

3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.

3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.

4. Prove that a central extension of an abelian group is nilpotent.

5. Give two different composition series for S_4 and show that they have the same composition factors.

6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A. Prove that rank(A/B) = rank(A) - rank(B). ADDED: I asserted in class, but did not prove, that a subgroup of a finitely generated abelian group is again finitely generated. As one class member pointed out, without this fact it's not even clear why it makes sense to talk about rank(B)! You may assume it (though it would have been better for me to have included proving this fact as part of the problem....)

7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then

[G: H_1 intersect H_2] <= [G:H_1][G:H_2].

Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.

Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.

(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)

8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.

9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.

## HOMEWORK 6 (due Oct 30)

1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.

2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then

dim V_X^G = number of orbits of X.

3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that

f(rho(g)(v)) = psi(g)(f(v))

for all g in G and all v in V.

Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by

H_0((V) = V^G.

(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)

5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)

6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.

7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.

7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.

7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.

## HOMEWORK 7 (due Nov 6)

1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)

2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.

2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.

2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.

3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W.

3a. Show that there is a unique linear transformation F satisfying

F(v tensor w) = f(v) tensor g(w)

for all v in V and all w in W. We denote this transformation by f tensor g.

3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?

4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form

v tensor w - w tensor v

for v,w in V.

Suppose dim V = n. What is dim Sym^2 V?

5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.

## HOMEWORK 8 (due Nov 13)

1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.

1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)

1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.

2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.

3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that

<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)

where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.

4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)

5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A.

5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.)

5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)

5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.

## HOMEWORK 9 (due Nov 20)

0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]

1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.

1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)

1b. Show that every Boolean ring is commutative.

2. Let phi : R -> S be a homomorphism of commutative rings with 1.

2a. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.

2b. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.

2c. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.

3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.

3a. Find (a generator of) Nil(Z/720Z).

3b. Show that Nil(R/Nil(R)) = 0.

3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.

3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.

4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.

4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.

5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.

6. Prove that the direct sum of any collection of free R-modules is also a free R-module.

7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.

7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]

7b. Show that M/N1 is a free Z-module.

7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]

7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.

7e. Use problem 5 to obtain a contradiction.

8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).

8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]

8b. Show that R is isomorphic to R^n for every positive integer n.