Math Studies Algebra II Prof. Clinton Conley Spring 2017 Contents 1 January 18, 2017 4 1.1 Logistics..................................................... 4 1.2 Modules..................................................... 4 2 January 20, 2017 5 2.1 Submodules................................................... 5 2.2 Module Homomorphisms............................................ 5 3 January 23, 2017 6 3.1 Generation of Modules............................................. 6 3.2 Free Modules.................................................. 6 4 January 25, 2017 7 4.1 Free Modules cont................................................ 7 5 January 27, 2017 8 5.1 Free Modules cont................................................ 8 5.2 Dimension.................................................... 9 6 January 30, 2017 9 6.1 Finitely Generated Modules over Principle Ideal Domains......................... 9 7 February 1, 2017 11 8 February 3, 2017 12 9 February 6, 2017 12 9.1 Finite Dimensional Vector Spaces....................................... 12 10 February 8, 2017 13 11 February 10, 2017 14 12 Feburary 13, 2017 15 12.1 Tensor Products................................................ 15 13 February 15, 2017 16 14 February 17, 2017 16 14.1 Exact Sequences................................................ 16 14.2 Splitting..................................................... 17 15 February 20, 2017 17 15.1 Topological Vector Spaces........................................... 17 16 February 22, 2017 18 1
17 February 24, 2017 18 18 February 27, 2017 19 18.1 Representation Theory............................................. 19 18.2 Decomposition of Representations...................................... 20 19 March 1, 2017 21 19.1 Maschke s Theorem............................................... 21 20 March 3, 2017 22 20.1 Artin-Wedderburn Theorem.......................................... 22 21 March 6, 2017 23 22 March 8, 2017 24 22.1 Examples Using C[G]............................................. 24 23 March 20, 2017 25 24 March 22, 2017 25 24.1 Extensions of Fields.............................................. 25 25 March 24, 2017 26 25.1 Extensions of Fields (cont.).......................................... 26 25.2 Straight Edge and Compass Constructions.................................. 26 26 March 27, 2017 27 26.1 Algebraic Extensions.............................................. 27 26.2 Splitting Fields................................................. 28 27 March 29, 2017 28 27.1 Algebraic Closures............................................... 28 28 March 31, 2017 29 28.1 Linear Orders.................................................. 29 28.2 Algebraic Closures (cont.)........................................... 29 29 April 3, 2017 30 29.1 Separable Polynomials............................................. 30 30 April 5, 2017 31 31 April 7, 2017 31 31.1 Galois Extensions................................................ 31 32 April 10, 2017 33 32.1 Galois Extensions (cont.)........................................... 33 33 April 12, 2017 34 33.1 Fun with Galois Theory............................................ 34 33.1.1 Normal Subgroups........................................... 34 33.1.2 Fundamental Theorem of Algebra.................................. 34 34 April 14, 2017 35 35 Radical Extensions 35 35.1 Solvable Groups................................................ 35 36 April 17, 2017 36 2
37 April 19, 2017 36 37.1 The Converse of Galois............................................. 36 38 April 24, 2017 37 39 April 26, 2017 38 39.1 Ultrafilters (cont.)............................................... 38 39.2 Some Combinatorics.............................................. 39 40 April 28, 2017 39 40.1 The Space of Ultrafilters............................................ 39 40.1.1 Crash Course in Topology....................................... 39 40.1.2 Topology on Space of Ultrafilters................................... 40 41 May 1, 2017 40 41.1 Semigroups................................................... 40 42 May 3, 2017 41 42.1 Minimal Subsemigroups............................................ 41 43 May 5, 2017 42 43.1 Hindman s Theorem.............................................. 42 3
1 January 18, 2017 1.1 Logistics Office: Wean 7121 Office Hours: Tuesday 1:30-3:00 Andrew ID: clintonc Grading: 20% homework (graded for completeness), 20% 2 midterms, 40% final 1.2 Modules Definition. A module is something for a ring to act on. Informally, modules are the equivalent of group actions for rings. Definition. In this course, a ring is not commutative and they don t necessarily have an identity. Definition. Let R = (R, + R, R ) be a ring with identity. A (left, unital) R-module is an abelian group M = (M, + M ) with an action : R M M such that: (a) (r + R s) m = r m + M s m (b) (r R s) m = r (s m) (c) r (m + M n) = r m + M r n (d) 1 m = m. Sometimes, item (d) is dropped. Example 1.1. 1. For any ring R, M = {0} with operation 0 + M 0 = 0 and R-action r 0 = 0 is an R-module. 2. For any ring R, M = R with the same operation + and action r m = rm is an R-module. Example 1.2. Let R = Z. If M is a unital Z-module, we have that More generally, for z Z, 1 m = m = 2 m = (1 + 1) m = m + m. z m = m + + m. }{{} z times Thus, the Z-action is determined completely by the group (M, + M ). In summary, abelian groups and Z-modules are the same thing. Example 1.3. Let R = R and consider the Euclidean plane R 2 = {(x, y) : x, y R}. Consider the R-action via r (x, y) = (rx, ry). This makes R 2 into an R-module. More generally, R-modules are exactly real vector spaces. In even more generality, if R is a field F, then F-modules are simply vector spaces over F. Example 1.4. Let R = F[x], the polynomial ring in one variable over a field F. Recall that F[x] is a Euclidean domain, and thus a PID, UFD, etc. Suppose we have an F[x]-module. This module is also an F-module, so we may as well view the module as an F-vector space V. We investigate x v for v V. We know that for α, β, w, v V. If we write x v as T (v), we have that x (αw + βv) = x (α w) + x (β v) T (αw + βv) = αt (w) + βt (v), which is exactly what it means for T to be linear. So, the map v x v is a linear transformation of the vector space V. Conversely, if V is an F-vector space and T : V V is a linear transformation, we get an F[x]-module structure on V via x v = T (v). Thus, F[x]-modules and F-vector spaces with a single designated linear transformation are equivalent notions. 4
Definition. Let R be a commutative ring with 1. An R-algebra is a (not necessarily commutative) ring A with 1 equipped with a unital (f(1) = 1) ring homomorphism f : R A wose image is in the center of A. Remark 1.5. On the homework, we will see that R-algebras are really special R-modules such that A is a ring R acts on A (which makes A a module) r (ab) = (r a)b = a(r b) in A. 2 January 20, 2017 Today, we discuss submodules and module homomorphisms. 2.1 Submodules Let R be a ring, M a R-module, and consider N M. We say that N is an R-submodule if N is an R-module with the same operations. Proposition 2.1 (Closure properties). N is an R-submodule if N is an abelian subgroup and R acts on N, i.e. for all r R and for all n N, r n N. Example 2.2. Let R = R, M = R 2 with the action begin scalar multiplication. 1. N 0 = {(x, x) : x R} is an R-submodule; it is indeed a subgroup and closed under the action. 2. N 1 = Q 2 is not an R-submodule; it is not closed under the action. 3. N 2 = {(x, y) : x = 0 y = 0} is not an R-submodule; it does not form a subgroup. Proposition 2.3 (Submodule criterion). Let M be a unital R-module and let N M. Then N is an R-submodule if and only if N for all r R and for all x, y N, x + r y N. 2.2 Module Homomorphisms Definition. Let R be a ring and let M, N be R-modules. A map ϕ : M N is an R-module homomorphism if it preserves the R-module structure. That is, ϕ is a group homomorphism from M to N ϕ preserves the R-action. Definition. An R-module isomorphism is a bijective R-module homomorphism. If there is an isomorphism between two R-modules M and N, we write M = N. Definition. The kernel of a function ϕ : M N for M, N R-modules is ker(ϕ) = {m M : ϕ(m) = 0 N }. Proposition 2.4. The kernel of an R-homomorphism is a submodule. Proposition 2.5. Let M be an R-module and let N be a R-submodule. Then, the quotient group M/N can be given an R-module structure by r (m + N) = r m + N. Upon doing so, the map ϕ : M M/N via ϕ(m) = m + N becomes an R-module homomorphism with ker(ϕ) = N. Definition. Suppose ϕ : M N is a function. Then, im(ϕ) = ϕ(m) = {ϕ(m) : m M}. Proposition 2.6. If ϕ : M N is an R-module homomorphism, then im(ϕ) is an R-submodule of N. 5
Theorem 2.7 (First Isomorphism Theorem for R-modules). If ϕ : M N is an R-module homomorphism, then M/ ker(ϕ) = im(ϕ). Definition. Let M, N be R-modules. Then, hom R (M, N) denotes the set of all R-module homomorphisms from M to N. Proposition 2.8. The set hom R (M, N) forms an abelian group under the operation (ϕ + ψ)(m) = ϕ(m) + ψ(m). Proposition 2.9. If R is commutative, the action (r ϕ)(m) = r [ϕ(m)] makes hom R (M, N) an R-module. 3 January 23, 2017 3.1 Generation of Modules For this section, assume that R is a ring with identity. Definition. Let M be a unital R-module and let A M. Then, RA is the R-submodule generated by A, the smallest R-submodule of M containing A. Formally, RA = N N S where S is the set of R-submodules of M containing A. Equivalently, Remark 3.1. Note that A RA because 1 R. RA = {r 0 a 0 + + r k a k : k N, r i R, a i A}. Remark 3.2. By convention, we set the empty sum to be 0. Definition. An R-submodule N M is finitely generated if there exists a finite A M such that N = RA. Definition. An R-submodule is cyclic if there exists an a M such that N = R{a} = {r a : r R} (we usually write Ra := R{a}). Remark 3.3. Cyclicity of Z-submodules coincides with the cyclicity of subgroups (abelian subgroups). Example 3.4. Consider the ring R = R and the R-module M = R 2. Then M is generated by A 0 = {(1, 0), (0, 1)}. M is also generated by A 1 = {(1, 1), (1, 1)}. 3.2 Free Modules Definition. Let M be an R-module. Then A M is said to freely generate M if for all x M there is a unique k N, unique and distinct a 0,..., a k A, and unique and nonzero r 0,..., r k R such that x = k i=0 r a. Proposition 3.5. Given a A, there is a canonical way to formally construct an R-module freely generated by A. Proof. Define the set F R (A) = {f : A R : {a A : f(a) 0 R } < }. Note that F R (A) is an abelian group via the operation (f + g)(a) = f(a) + g(a). Let R act on F R (A) by (r f)(a) = r(f(a)). It is easy to see that this makes F R (A) into an R-module. We identify a copy of A inside F R (A) via associating a A with f a F R (A) where { 1 R b = a f a (b) = 0 R b a. (1) Then, f F R (A) has the form f(a 0 ) f a0 + + f(a k ) f ak where a 0,..., a k list the elements of f(a i ) such that f(a i ) 0. 6
4 January 25, 2017 4.1 Free Modules cont. Remark 4.1. If A is finite, say A = n, then F R (A) = {f : A R} = R n with coordinatewise operations. To check whether the same generating set A freely generates M, it is enough to check that whenever a 0,..., a k A are distinct and r 0,..., r k such that k r i a i = 0, then r 0 = = r k = 0. i=0 Whether some set A generates M depends critically upon the acting ring. For instance, let M = R 2. Then, 1. As an R-module, M is 2-generated. 2. As a Q-module, M is not finitely (not even countably) generated. Theorem 4.2 (Universal Property of Free R-modules). Suppose A is some set, R is a ring with 1, M is some R-module, and g : A M is some function. Then there exists a unique R-module homomorphism γ : F R (A) M extending g, i.e. g(a) = γ(a) for all a A. Moreover, where B = im(g). Proof. We first prove existence and then uniqueness. Existence im(γ) = RB For f F R (A), define γ(f) = a A f(a) g(a) A, which is a finite sum of nonzero terms. Note that γ is indeed an R-module homomorphism since and γ(r f) = a A Then, γ indeed extends g since Uniqueness γ(f 0 + f 1 ) = a A(f 0 (a) + f 1 (a)) g(a) (rf(a)) g(a) = a A = f 0 (a) g(a) + f 1 (a) g(a) a A a A = γ(f 0 ) + γ(f 1 ) r (f(a) g(a)) = r f(a) g(a) = r γ(a). γ(f a ) = 1 g(a). Let γ 0, γ 1 : F R (A) M be R-module homomorphisms extending g and consider the new function δ = γ 0 γ 1, which is also an R-module homomorphism δ : F R (A) M. Note that for all a A, δ(a) = γ 0 (a) γ 1 (a) = g(a) g(a) = 0, i.e. ker(δ) is an R-submodule of F R (A) which contains A. Thus, ker(γ) = F R (A) and thus δ = 0 and we conclude as desired. a A 7