MTH 419H
Honors Algebra II

1. Geometry in n-space (partly in 4.5)
2. Determinants, eigenvectors, diagonalization of matrices, Jordan form (1.3, 4.3--4.6)
3. Structure of groups (6.1--6.5)
4. Ring theory (10.3--10.6, 11.5)
5. Modules, abelian groups, Jordan form (12.4--12.7)
6. Field theory, ruler-and-compass constructions (13.1--13.9)
7. Galois theory, solving equations by radicals(14.1--14.9)

• Geometry in n-space, equations and parametrizations for linear subspaces and polyhedra
• Determinants: definitions from geometry, algebraic properties, explicit formula (sum over permutations), and recursion (minor expansion); inverse matrices; det(AB) = det(A) det(B); computations via row reduction.
• Diagonalization, characteristic polynomial, eigenvectors, discriminant
• Jordan form, generalized eigenspace, nilpotent maps, Cayley-Hamilton Theorem

1. (a)In R⁴, consider the vectors v₀ = (1,1,1,1) , v₁ = (1,2,3,4) , v₂ = (4,3,2,1) . Define two 2-dimensional planes by:H₁={ x  |  x•v₁= x•v₂ = 1 }
   H₂={ x = v₀+λ₁v₁+λ₂v₂  |  λ₁,λ₂ ∈ R } Here H₁ is defined by dot-product equations, whereas H₂ is parametrized by (λ₁,λ₂).(These are the two dual forms for writing any k-dimensional plane in Rⁿ.)   Write each of these planes in the other form.Hint: First solve a system of linear equations to find two vectors orthogonal to both v₁ and v₂ .
   (b) Write both of the above forms for the unique 2-dimensional plane H₃ which contains the three points v₀ , v₁ , v₂ . Hint: Find two directions which point along H₃ .
2. Recall the definition of a half-space of Rⁿ: H_v^c+ = { x  |  x•v ≥ c } .Now, an n-dimensional polytope P is the intersection of several half-spaces, assuming that the intersection is non-empty and none of the inequalities is redundant.A facet is the intersection of P with one of the (n-1)-dimensional planes H_v^c= { x  |  x•v = c },and a k-dimensional face is a non-empty intersection of (n-k) facets, assuming none of the (n-k) equalities is redundant.
   Find the number of k-dimensional faces of the n-dimensional cube Cub_n , and of the n-dimensional simplex Δ_n .
3. Consider the n-dimensional simplexΔ(0,e₁,...,e_n) , the convex hull of the origin and the standard basis vectors e₁ ,..., e_n . Prove that the volume of this simplex is 1/n! . Hint: Parametrize the points of the simplex with parameters x₁ ,..., x_n , and evaluate an iterated n-fold integral.
   Extra Credit: Investigate the volume of the n-dimensional ball B_n. Find a neat formula for its volume, depending on n (distinguish n even vs odd).
4. Extra Credit: Reprise of an exercise from last semester. Recall the 8-element group of quaternion units Q = { ±1, ±i, ±j, ±k } inside the ring H of quaternions. (See the 418H Final Review HW, Ex.4.)
   Use the representation of H as 2 × 2 complex matrices to find a 4-dimensional polytope P in R⁴ whose symmetry group is Q. Namely, take some random point p = (a,b) in C² , and take 8 points g•p for g ∈Q (that is, multiply the matrix g by the column vector p). Consider these 8 points as points in R⁴ by taking real and imaginary parts of the complex coordinates, and take these 8 points p₁,...,p₈ as the vertices of your polyhedron. The rest is gotten by taking the convex hull (weighted averages) of these points : that is, all points v = c₁p₁ + ... + c₈p₈ , where the coefficients run over all c₁,...,c₈ ≥ 0 with c₁ +... + c₈ = 1 .
   Figure out all the faces of this polyhedron.One strategy is to take any 4 vertices and find the hyperplane H_v^c which they span. If all other vertices lie on one side of this hyperplane (i.e. in one of the half-spaces), then the simplex spanned by the 4 vertices is a facet. Now consider all non-empty, non-redundant intersections of facets.

1. (a) We say that points v₁,...,v_k ∈ Rⁿ are affinely independent if no point is in the affine space spanned by the others. Give an algebraic definition of affine independence.
   (b) Let v'_i = (v,1) ∈ Rⁿ⁺¹ , the vector in one higher dimension with the last coordinate fixed to be 1. Show that v₁,...,v_k are affinely independent if and only if v'₁,...,v'_k are linearly independent.
2. In class, we defined the determinant using 4 weak properties (i')--(iv'), and alternatively by 4 stronger properties (i)--(iv). Prove that these two sets of properties are equivalent: any function satisfying one satisfies the other.
3. Use the defining properties (i)--(iv) to prove that det(A*B) = det(A) det(B), where * means matrix multiplication. Hint: Consider the function f : X → det(A*X), where X = (v₁,...,v_n) is a matrix of n column vectors. Which of the properties (i)--(iv) does f(X) satisfy?
4. Use the explicit formula definition of det(A) to show that det(A) = det(A^t) , where A^t denotes the transposed matrix (flipped across the main diagonal). You should use the substitution σ = π^-1 in the summation formula.
5. Evaluate the determinant of a general 2 × 2 matrix directly by the geometric definiton: that is, find area via a double integral.
6. Extra Credit: Let x₁,...,x_n be constants, and let A = (a_ij) be the matrix with entries a_ij = x_j^i-1 . Thus, each column is of the form 1, x, x²,..., x^n-1 . Find a simple formula for the determinant det(A). Hint: Do some convenient column-reduction

1. Given an n × n matrix A = (a_ij) , the (i,j) minor of A is the (n-1) × (n-1) matrix A_ij obtained by removing the i^th row and j^th column. The minor expansion of det(A) along the i^th row is the formula:

   det(A)   =   Σ_j=1ⁿ   a_ij det(A_ij) .

   Let D(A) denote the right-hand side of this equation.
   1. Prove the formula by showing that D(A) satisfies the weak properties (i')--(iv') which characterize det(A). The only slightly difficult one is (ii'): if two columns of A are identical, then D(A) = 0 . Use the fact that if columns r and s of A are identical, then det(A_ir) = (-1)^r+s+1det(A_is) . (Why?)
   2. Now prove the minor expansion formula using the permutation formula for det(A). Modify this as follows: suppose a matrix A has its rows and columns labelled not by [n] := {1,2,...,n}, but rather by some other ordered n-element sets I and J. (For example, I = J = {red, green, blue}, so that we could refer to the red row and green column, etc.) Now let S(I,J) be the set of all bijections (one-to-one and onto maps) π : I → J . By relabelling, these bijections correspond immediately to permutations of [n] . Then we have:

      det(A)   =   Σ_π∈S(I,J)   (-1)^π a_i1,π(i1)...a_in,π(in) ,

      where I = {i₁, ... , i_n} .
      Expand D(A) using this formula, labelling the rows and columns of the minor A_ij by the sets I = [n] - {i} and J = [n] - {j} . Show that this reduces to the formula for det(A) .
2. Extra Credit: In class we showed the formula: A A^~ = det(A) I , where we use the adjoint matrix A^~ = (a_ij^~) = ( (-1)^i+j det(A_ji) ) . This formula holds for a matrix A with entries in any commutative ring R ; that is, A ∈ M_n×n(R) . Check that all our proofs work in this generality.
   Use this formula to determine the group of matrices A ∈ M_n×n(R) having an inverse in M_n×n(R) .
   Now let R be the integers Z or the polynomial ring Q[x], and show there are lots of matrices which are invertible in M_n×n(R) . Hint: it is easy to find the determinant of an upper-triangular matrix. Now multiply.

1. Proposition: If T : V → V is a linear transformation on a vector space V, and v₁,...,v_m are eigenvectors with distinct eigenvalues, T(v_i) = λ_iv_i with λ_i ≠ λ_j , then v₁,...,v_m are linearly independent.
   1. Prove this by induction on m. If there were a non-trivial relation, we could write v_m = a₁v₁+...+a_m-1v_m-1, and apply T.
   2. Reconstruct the proof I gave in class, which involves writing the equationsλ₁^j a₁ v₁ + ... + λ_m^j a_m v_m = 0for j = 0, 1, ... , m-1 in terms of the matrix of column vectors [a₁v₁,...,a_mv_m], the Vandermonde matrix L = (λ_i^j), and the zero matrix.The key fact is the Vandermonde determinant: det(L) = Π_{i < j} (λ_i - λ_j) , which implies L is invertible.
2. Invariants of a linear transformation. Let T : V → V be a linear transformation on a vector space V. We can define a function f(T) by choosing a basis B, writing the matrix A = [T]_B^B, and giving some expression in the coefficients of A = (a_ij), for example f(T) = a₁₁. However, a different choice of basis would give a different answer for f(T) , so this is really a function of T and the basis B.
   We are interested in functions of A which depend only on T, giving the same answer for any choice of basis. Such functions are called invariants.
   1. Show that for a given T, the determinant det([T]_B^B) is the same for any choice of basis B. Thus, we can unambiguously write det(T) without specifying a basis. Hint: What is the matrix operation for change-of-basis?
   2. The trace of a matrix A = (a_ij) is the sum of the diagonal entries: tr(A) := a_ij + ... + a_nn . Show directly that tr(A*B) = tr(B*A), where * means matrix multiplcation. Conclude that for a given T, the trace tr([T]_B^B) is the same for any choice of basis.
   3. Are there any other invariants besides det and tr? For n = 2 essentially not (except trivial modifications like det² + 3 tr), but for larger n there are indeed! You might try finding some before looking at the rest of the problem here.
3. An n × n matrix N is nilpotent if some power of N vanishes: N^k = 0.
   1. Show that any upper triangular matrix A is nilpotent: that is, A = (a_ij) with zero entries on and below the main diagonal, a_ij = 0 for i ≥ j .
   2. Show that if N is nilpotent, then its only eigenvalue is λ = 0.
   3. Extra Credit: Show that N is nilpotent if and only if the characteristic polynomial of N is det(N-xI) = xⁿ. You might need to assume that N can be put in Jordan form.
4. Extra Credit: We stated in class that there is a function P(A), a polynomial in the entries of the n × n matrix A, such that P(A) = 0 if and only if A has n distinct eigenvalues. This is called the discriminant, often denoted P(A) = disc(A). We computed that for n = 2,disc(A) = (a₁₁ + a₂₂)² -4(a₁₁a₂₂ - a₁₂a₂₁) .Express this quantity in terms of the invariants of problem 1. Give an argument why disc(A) is an invariant for any n. Explicitly find disc(A) for n = 3. Explicitly find disc(A) in terms of the eigenvalues of A, if A is diagonalizable. Say whatever else you can about disc(A). References: there is a little in Artin Ch 14.3, p.548, and much more in Serge Lang's Algebra, 3rd ed., Ch IV.8.

Algorithm to compute the Jordan form of a transformation T on Cⁿ, which has standard basis e = {e₁,...,e_n}. Preliminaries:

• All vectors will be denoted as n×1 column vectors in standard coordinates.
• A subspace W ⊂ Cⁿ is always specified by a basis w = {w₁,...,w_m}, which can also be considered as an n×m matrix of column vectors. Using column reduction, we may require that some submatrix of w consisting of rows (i₁,...,i_m) forms an m×m identity matrix.
• If T(W) ⊂ W, we say W is a T-invariant subspace, and we let T|_W denote the restricted transformation T acting on W . Below we will find the invariant subspaces W_i = Span(v_i,1,...,v_i,ni) corresponding to the Jordan blocks.
• Procedure: Given a T-invariant subspace W, compute the the nullspace Ker(T|_W). Form the n×m matrix of column vectors M = [T(w₁),...,T(w_m)].Perform row reduction on M to find its kernel vectors u₁,...,u_t ∈ C^m, then matrix multiply w*u_i to get a basis of Ker(T|_W).
  For extra efficiency, we could instead work with the m×m matrix M' consisting of rows (i₁,...,i_m) of M. The vectors u ∈ Ker(M') ⊂ C^m are the same as those for M, and we again take w*u.
• Procedure: Given two spaces W ⊂ V ⊂ Cⁿ, find a complementary space W' such that V = W ⊕ W' ; that is, V = Span(W,W'), and W ∩ W' = 0. Letting m = dim(W), p = dim(V), form the n × (m+p) matrix of the basis vectors of W followed by the basis vectors of V :   M := [w₁,...,w_m,v₁,...,v_p]. Reduce the last p columns by adding multiples of the first m columns (for efficiency, concentrate on rows (i₁,...,i_m) ). Finally, column-reduce the last p columns, obtaining p-m non-zero column vectors, a basis of W'.

Now, given T, we compute the basis vectors v_i,j.

• First find the eigenvalues of T, the roots x = λ_k of the characteristic polynomial f(x) = det(M-xI) . The rational root test might be useful here, as well as other methods of solving polynomials.
• For each distinct λ_k , use row reduction to compute Ker(T-λ_kI)^q for q = 1,2,..., which increases until we reach a q with: Ker (T-λ_kI)^q-1 ≠ Ker (T-λ_kI)^q = Ker (T-λ_kI)^q+1 .Let V_k := Ker (T-λ_kI)^q , the generalized eigenspace of T corresponding to all the Jordan blocks with the eigenvalue λ_k .We thus have a decomposition V = ⊕ V_k into T-invariant subspaces.
• Let T_k := (T-λ_kI)|_Vk, a nilpotent transformation on V_k with T_k^q = 0 but T_k^q-1 ≠ 0 .We want to find a Jordan basis for this T_k . First, we find a complement V_k,1 to Ker T_k^q-1 ⊂ Ker T_k^q = V_k . We repeat this, finding complementary spaces V_k,j so that:V_k = Ker T_k^q = Ker T_k^q-1 ⊕ V_k,1
  Ker T_k^q-1 = Ker T_k^q-2 ⊕ T_k(V_k,1) ⊕ V_k,2
  Ker T_k^q-2 = Ker T_k^q-3 ⊕ T_k²(V_k,1) ⊕ T_k(V_k,2) ⊕ V_k,3
  ...We get a decomposition V_k = ⊕ T_k^p(V_k,j), summed over all j and p.
• Now we take v_1,q to be a basis vector of V_1,1, and v_1,q-p := T₁^pv_1,q. We repeat this with another basis vector v_2,q of V_1,1, until we fill out V_1,1. We go on with a basis vector v_i,q-1 of V_1,2, and continue in this way until we have a basis of V₁ .Repeating with V₂ , V₃ , ... produces the Jordan basis.

1. Show that if we let D and N be the diagonal and off-diagonal parts of J (respectively), then we get the decomposition M = P D P^-1 + P N P^-1, where the first term is diagonalizable, the second term is nilpotent (some power equals zero), and the two summands commute with each other under matrix multiplication.
2. Extra Credit: Give another proof of the Jordan Form Theorem by showing that the vectors v_i,j found in the algorithm are actually a basis of V. The main point is that the spaces V_k are linearly independent and span V, and the spaces T_k^p(V_k,j) are linearly independent and span V_k .
3. Consider the following matrices J, P, and M = PJP^-1, respectively:

   2	1	0		1	0	0		0	2	-1
   0	2	0		1	1	1		-2	4	-1
   0	0	2		0	1	2		0	0	2

   Use the above (improved) algorithm to recover the Jordan form J of M, along with the Jordan basis v_i,j given by the columns of P.

1. Find H_a , the n-dimensional affine hyperplane containing Δ. (Define H_a by equations and by parametrizing.)   Translate H_a to a linear hyperplane H₀ by subtracting the vector v = (e₁+...+e_n+1) / n , and describe this plane similarly.
2. Consider the following basis of H₀:B = {e₁ - e_n+1 , ..., e_n - e_n+1} .Find the volume of the n-dimensional parallelopiped Cub(B) spanned by the vectors of B.
   Hint: Extend B to a basis B' of Rⁿ⁺¹ by including the orthonormal vector v' := (e₁+...+e_n+1) / √(n+1) . The n-volume of Cub(B) is the same as the (n+1)-volume of Cub(B').
3. The simplex Δ₀ = Δ - v lies inside H₀ and is spanned by the vectors e₁ - v ,..., e_n+1 - v. Find the n-volume of Δ₀.
   Hint: vol(Δ₀) = vol(Cub(B)) / n! . Why?
4. Show that any (n+1)×(n+1) permutation matrix takes H₀ to itself, and is in fact a symmetry of Δ₀. Thus, Δ₀ is a regular (maximally symmetric) n-simplex.
5. Show that permutations are the only symmetries of Δ as a rigid object (i.e., symmetries preserving the distance between points).
6. Show that even permutations (products of an even number of transpositions) act by det = 1 matrices (orientation-preseving), and odd permutations act by det = -1 matrices (orientation-reversing). Thus, the oriented symmetries form the alternating group A_n, the subgroup of even permutations, which is a normal subgroup of S_n by Prob 1 of HW 2/14.
7. Extra Credit: Find an orthonormal basis O = {u₁,...,u_n} of H₀. That is, the dot product u_i•u_j = 1 if i = j, and =0 otherwise.
   Hint: Start with an arbitrary basis B = {v₁,...,v_n} of H₀.Normalize u_n := v₁ / |v₁| . Subtract the v₁-component (v₁•v₂) v₁ from v₂, then normalize, getting: u₂ := v / |v| ,   where  v = v₂ - (v₁•v₂ . Continue similarly for all u_i.
   Having an orthonormal basis O is tantamount to identifying H₀ with Rⁿ as a metric space (vector space with dot-product).
8. Extra Credit:The punchline is to explicitly determine the vertices of a regular n-simplex in Rⁿ. For simplicity, try n = 3.
   Writing the vertices of Δ₀ in the basis O gives coordinates for a regular simplex in Rⁿ.Could you describe the matrices by which permutations act?

1. Consider the group SL_n(F) := {A ∈ M_n(F) | det(A) = 1}. Show that SL_n is a normal subgroup of GL_n.
   Show that if G ⊂ GL_n is any group of matrices, then G ∩ SL_n is a normal subgroup of G. For example, the group O(n) ∩ SL_n is denoted SO(n), a connected component of O(n). Also, if the permutation group S_n is realized as the symmetries of an (n-1)-dimensional simplex, then the det=1 symmetries form the alternating group A_n := S_n ∩ SL_n, a normal subgroup of S_n.
2. For G a group, we define the center Z(G) := {z ∈ G | gz = zg for all g ∈ G}. Let G be the general linear group GL_n(F), where F is a field, and show that Z(G) is the set of invertible scalar matrices λI where λ ∈ F - {0}.
   Hint: Let X be an unknown matrix, and solve the system of equations X (I + E_ij) = (I + E_ij) X , where E_ij is the matrix with 1 in position (i,j) and 0 elsewhere.
3. For g ∈ G, define the centralizer of g by Z(g) := {z ∈ G | gz = zg}. Show that for x,y,g ∈ G, we have xgx^-1 = ygy^-1 if and only if x ≡ y mod Z(g) ; that is, y = xz for some z ∈ Z(g). Letting C(g) = {xgx^-1 | x ∈ G}, the conjugacy class of g, conclude that |C(g)| = |G| / |Z(g)|.
4. Prove: If G is a group of order |G| = p^k for k > 0, then the center Z(G) is non-trivial, |Z(G)| > 1.
   Hint: Use the class equation |G| = |Z(G)| + Σ_g |G| / |Z(g)| , where the summation is over conjugacy classes C(g) with Z(g) ≠ G.
5. In class, we showed that any group of order |G| = p² is abelian.
   Now prove that G is isomorphic to the cyclic group C_p² or the product of cyclic groups C_p × C_p.
   Hint: C_p × C_p is generated by two commuting elements of order p.

1. Recall the definition of a group G acting on a set S. Our main examples are:
   • The first-order left-multiplication action of G on S = G itself by g*s = gs (multiply in the group).
   • The first-order conjugation action of G on S = G itself by g*s = gsg^-1 .
   • The second-order actions derived from each of the above, of G on S = { all subsets S ⊂ G } by g*S := { g*s for s ∈ S }.
   Our proofs of the Sylow Theorems depend on variations of these actions.
   Consider the dihedral group G = D₃, the symmetry group of the triangle. Follow the proofs of the Sylow Theorems for this special case and p = 2 or 3, by explicitly examining:
   1. G acting by left multiplication on S = { all subsets of size p^e } . Find an orbit O_S whose order is prime to p. Then H = Stab_G(S) is a Sylow p-subgroup.
   2. Given a subgroup K ⊂ G and a Sylow p-subgroup H ⊂ G, consider K acting by left multiplication on S = { gH for g ∈ G } = G/H, the set of H-cosets of G. Consider an orbit O_gH whose order is prime to p, and see that gHg^-1 ∪ K ⊂ K is a Sylow p-subgroup of K.
   3. G acting by conjugation on S = { all Sylow p-subgroups of G } . Then S is a single orbit O_H, where H ∈ S. Then #S = #G / #N_G(H) divides #G / #H = m.
   4. A particular Sylow p-subgroup H acting by conjugation on S = { all Sylow p-subgroups of G }. Then all orbits have order divisible by p except for O_H = {H}. Thus #S ≡ 1 mod p.
2. Extra Credit: Artin Ex Ch 6.4 #6, 11, pp 232. Sylow p-subgroups of GL_n(Z_p) .

1. Ex Ch 6.4 #14, 18, p. 232. Show that the only simple groups of order ≤ 60 are the cyclic groups C_p for p prime and the alternating group A₅.
2. Recall from Artin Ch 6, Prop. 4.9(b), p 207, that there is at most one non-abelian group of order 21: if it exists, it must be generated by two elements x,y which satisfy the relations x⁷ = y³ = 1 and yx = yx². Show this group exists in two ways.
   • Check the required properties for a certain subgroup of GL₂(Z₇) given in Artin Ex Ch 6.4 #4, p 231. Of course, this does not explain how Artin found this subgroup!
   • Construct the group in a more "natural" way by considering the finitely presented group G := Pres< x,y | x⁷ , y³ , yxy^-1x^-2 > .This group by definition has the required generators and relations, but it is not clear that it has order 21 or even that it is finite. Indeed, G might be too small, meaining the relations contradict each other, or G might be too large, meaning the relations are incomplete. Check that |G| = 21 by constructing the Cayley graph of G : draw a directed graph with one x-colored vertex entering each vertex and one such leaving, and the same for y-colored edges. Starting from the identity vertex, each edge goes to a new vertex unless a relation forces it to be identified with an old vertex. Continue until no new vertices are possible, and check that you have 21 vertices altogether.Alternatively, do all this using a table as described in Artin Ch 6.9, p. 224.
3. Ex Ch 6.7 #3. Cyclic words give conjugacy classes of the free group.
4. The commutator of two group elements x,y ∈ G is defined as: [x,y] := xyx^-1y^-1 . Show the following:
   1. [x,y] = 1 if and only if x and y commute.
   2. Let [G,G] := { [x₁,y₁][x₂,y₂]...  |  x_i,y_i ∈ G } , the set of all products of commutators. (Thanks to James Wolter for the correction.) Then [G,G] is a normal subgroup of G and G / [G,G] is abelian.
   3. Show that if H is a normal subgroup of G with G/H abelian, then [G,G] ⊂ H . Thus there is a natural surjective map G / [G,G] → G / H . That is, G / [G,G] is the largest abelian quotient of G.
   4. If G is a non-abelian group which is simple (no non-trivial normal subgroups), then [G,G] = G.
   5. Determine [G,G] and G / [G,G] for some familiar groups such as C_n, D_n, the quaternion 8-group, S₄, S_n for n ≥ 5, the free group Free(n).
5. Consider the iterated commutator: G₀ := G ,   G₁ := [G₀,G₀] ,   G₂ := [G₁,G₁] , etc. We say a group G is solvable if eventually some G_k = {1}. (This will later turn out to have an amazing connection with solvablity of polynomial equations, generalizing the quadratic formula.)
   1. Let GL_n(F) be the invertible matrices over any field F . Let H be the subgroup of all upper-triangular matrices with non-zero diagonal entries. Show that H is a solvable group.
   2. Determine whether our familiar groups are solvable.Give an example of a non-solvable group.

1. Let R be a commutative ring.
   • An ideal K ⊂ R is a subset with: k₁, k₂ ∈ K  ⇒  k₁ - k₂ ∈ K          and         r ∈ R ,   k ∈ K  ⇒ r.k ∈ K .For any elements r₁,...,r_n, we have the ideal (r₁,...,r_n) := { a₁r₁ + ... + a_nr_n  |  a_i ∈ R } .
   • The quotient ring R/K is the set of all additive cosets { r + K  |  r ∈ R } with the operations: (r₁ + K) + (r₂ + K) := (r₁ + r₂) + K         and         (r₁ + K).(r₂ + K) := (r₁.r₂) + K .We write r ≡ r' mod K to mean r + K = r' + K , or equivalently r - r' ∈ K .
   Prove the basic facts about quotient rings.
   1. R/K is a well-defined commutative ring: that is, if r₁ ≡ r'₁ and r₂ ≡ r'₂ mod K , then r₁ + r₂ ≡ r'₁ + r'₂ and r₁ . r₂ ≡ r'₁ . r'₂ .
   2. There is a natural onto homomorphism π : R → R/K defined by π(r) = r + K . We have Ker(π) = K .
   3. The homomorphism π gives a one-to-one correspondence between the ideals J^~ ⊂ R/K and the ideals J ⊂ R with K ⊂ J .
   4. The commutative ring R is a field if and only if the only ideals of R are K = (0) and K = R .
   5. The quotient R/K is a field if and only if K is a maximal ideal : that is, if there is no ideal larger than K but smaller than all of R.
2. Consider a quotient R = F[x] / (f(x)) , where F is a field and f(x) is a polynomial of degree n .
   • The standard form of a polynomial a(x) is its remainder under division by f(x): that is, we write a(x) = q(x)f(x) + r(x) , where deg(r) < n , and note that a(x) ≡ r(x) mod (f(x)).
   • To add, subtract, or multiply standard forms, perform the operation in F[x], then reduce to standard form.
   • If f(x) is an irreducible polynomial, then (f(x)) is a maximal ideal in F[x], so R is field and we can perform division. Namely, given a(x) not in (f(x)), use the Euclidean algorithm to write m(x)a(x) + n(x)f(x) = 1 , then write 1/a(x) ≡ m(x) mod f(x) .
   Practice computing the four field operations for:
   1. Q[x]/(x³ + x + 1)
   2. F₄ := F₂[x]/(x² + x + 1), where F₂ = Z₂ , the field with 2 elements. Make addition and multiplication tables, and compute 1/a for each a ∈ F₄ .
   3. Extra Credit: Find an irreducible polynomial of degree 3 over F₂, and construct the field with 8 elements. Do the above exercises for this field.
3. Extra Credit: Let R be a non-commutative ring. Suppose that a,b ∈ R are elements such that (1 - ab) is invertible in R. Show that (1 - ba) is also invertible in R .
   Hint: Imagine you could expand (1 - ab)^-1 as an infinite geometric series. How to modify it to get (1 - ba)^-1 ? This does not constitute a proof, but it does suggest an answer.

1. We saw that the multiplicative group is (F_q^×, •) = C_q-1 , always a cyclic group.Determine the type of the additive group (F_q , +).
2. Define the Frobenius mapping   Frob : F_q → F_q   by   Frob(a) = a^p .
   1. Prove that Frob is a ring homomorphism.Hint: Apply the Binomial Theorem.
   2. Prove that Frob is an isomorphism. Hint: What could be its kernel?
   3. Prove that every element a ∈ F_q has a p^th root: an element b ∈ F_q with b^p = a .
3. Artin p 533, Ch 13.6 #4. For each element of F₈ , determine the irreducible polynomial in F₂[x] which it satisfies. For example an element a ∈ F₂ ⊂ F₈ satisfies the polynomial f(x) = x - a . The other elements will satisfy degree 3 polynomials. Given β ∈ F₈, there is an F₂-linear combination of 1, β, β², β³ which is zero.
4. Artin p 533, Ch 13.6 #15. Both   F₂[x]/(x³ + x² + 1)   and   F₂[x]/(x³ + x + 1)   are fields of order 8. Find the explicit isomorphism between them: that is, give the image of each element of the first field in the second. Hint: Solve the first polynomial in the second field.
5. Prove uniqueness for fields of prime order p . That is, if F is a field of order p, then F is isomorphic to F_p = Z/(p), the clock-arithmetic field of integers modulo p. Give an explicit isomorphism.