Archived version (no quizzes, exams, or solutions; minor cleaning) Instructor: Anna Medvedovsky Syllabus Announcements More on optional projects. Older announcements Resources Homework assignments Post date Due date HW assignment 4/17 Final 4/24/23 5/3 HW #7 4/2 Final 4/7/23 4/14 HW #6 3/17 Final 3/25/23 3/31 HW #5 2/27 Edited 3/5/23 3/17 HW #4 2/13 Final 2/17/23 2/27 HW #3 1/27 Final 2/3/23 2/13 HW #2 1/20 Final 1/23/23 1/27 HW #1 Class meetings Date Class Topics W 5/3 41 Abel-Ruffini theorem: A general polynomial of degree ≥ 5 over a field of characteristic zero is not solvable by radicals. Reminder about simple grouops, composition series, Jordan-Hölder theorem, solvable groups. Kummer theory: if K is a field of characteristic zero containing all pth roots of unity, then Galois extensions L of K with Galois group Zp are the same as radical extensions of K. Modulo mucking aroun with roots of unity, the Galois correspondence reduces the Abel-Ruffini theorem to the statement that a polynomial equation over a field of characteristic zero is solvable by radicals if and only if its Galois group is a solvable group. Since Sn not solvable for n ≥ 5, and since “most” polynomials of degree n have Galois group Sn, “most” polynomials of degree degree ≥ 5 are not solvable by radicals. M 5/1 40 End of the proof of the Galois correspondence: conjugate subfields correspond to conjugate subgroups of the Galois group, so that subfields normal over the base correspond to normal subgroups; and in this case the Galois group of the subfield over the base is realized by the corresponding quotient group. A very little bit about cyclotomic extensions of Q. For p prime, Gal(Q(ζp)/Q) is isomorphic to Zp×, where the automorphism ζp → ζpa is mapped to [a] in Zp×. (This is true if p is not prime as well, but it relies on proving that the general nth cyclotomic polynomial Φn(x):= ∏1 ≤ a ≤ n, gcd(a, n) = 1(x – ζna) is in Z[x] (easy) and irreducible (trickier, see BB Theorem 8.5.3).) Course evaluations. F 4/28 39 Proof of the Galois correspondence: given L Galois over K with G = Gal(L/K) we want to show that (a) for any intermediate extension E we have E = LGal(L/E) and (b) for any subgroup H of G we have H = Gal(L/LH). Containments ⊆ in each case follow from the definitions (and in fact require no asssumptions on L over K). The full equality for (a) follows from the two ⊆ containments for E and H := Gal(L/E) together with degree equalities for the Galois extensions L over E and L over LH that follow from the corollary to the embedding extension theorem. Finally, the full equality from (b) comes constructing an explicit polynomial of degree #H and with coefficients in LH satisfied by a primitive element of the extension L over LH. W 4/26 38 Galois correspondence for Q(ζ8)/Q. Presentation and subgroup diagram for D4. Group work: Galois correspondence for Q(i, 21/4)/Q. (Hint: The Galois groups is D4, and we can take for generators r and f, where r is the order-4 automorphism that maps 21/4 to 21/4i and fixes i, and f is the order-2 automorphism that fixes 21/4 and maps i to –i. The field Q(21/4 ζ8) will play a role.) M 4/24 37 The set maps between intermediate subextension of a fixed extension L/K and subgroups of Aut(L/K) definied last time are inclusion reversing. Statement of the Galois correspondence (aka fundamental theorem of Galois theory): if L is Galois over K, then these set maps are inverses of each other, so that we get a one-to-one inclusion-reversing correspondence between the set of extensions of K contained in L and the set of subgroups of Gal(L/K). Moreover, intermediate extensions E normal over K correspond to normal subgroups of Gal(L/K), and in that case restriction to E induces an isomorphism of the quotient with Gal(E/K). The Galois group of a separable polynomial f over K of degree n (that is, the group Gal(L/K), where L is a splitting field for f) acts faithfully on the roots of f and hence embeds into Sn. If f is irreducible over K, then the action is transitive. Example: Galois correspondence for Gal(Q(21/3, ω)/Q) F 4/21 36 End of proof of equivalent definitions of normal extensions: finish (b) ⇒ (c) and (c) ⇒ (a). Corollary: for finite L/K, we have #Aut(L/K) = [L:K] if and only if L/K is “Galois”: both normal and separable. Towards the Galois correspondence: Given an extension L/K, we define a set map from the set of intermediate extensions (fields E with K ⊆ E ⊆ L; note that if L over K is Galois, then so is L over E) to the set of subgroups of Aut(L/K) by mapping E to Aut(L/E). We also have a map back from subgroups H of Aut(L/K) to intermediate extensions by sending H to the fixed field LH. W 4/19 35 Primitive element theorem (BB Theorem 8.2.8, but end of proof as in K. Conrad’s notes on separability): any finite separable extension is simple. Example of a finite extension with no primitive element. Normal extensions: finite L/K is normal if any of the following equivalent conditions are satisfied: (a) L is a splitting field for a polynomial in K[x], (b) all extensions to L of an embedding of K into a field F have the same image in F, (c) any irreducible polynomial in K[x] that has a root in L splits completely in L. We proved (a) ⇒ (b) and started on (b) ⇒ (c). Note that the equivalence (a) ⇔ (b) is the last part of the embedding extension theorem. F 4/14 34 Consequences of the embedding extension theorem for separability. See Theorem 3.5 and Theorem 3.13 in K. Conrad’s notes on separability. The extension K(α1, ..., αn) over K is separable if and only if the elements α1, ..., αn are all separable over K. In a tower M/L/K, the extension M/K is separable if and only if both M/L and L/K are separable extensions. Primitive element theorem (next time). Consequences of separability for finite fields. See BB section 6.5. There’s a field of pn elements for every prime p and every n ≥ 1 (the set of roots of fn(x) = xpn – x in any splitting field of fn over Fp has pn distinct elements and forms a field). Corollary: Fp[x] has at least one irreducible polynomial of every degree n ≥ 1. Fpn contains a copy of Fpm if and only if m divides n (one direction follows from multiplicativity of degrees in towers; for the other, if m divides n, then fm(x) divides fn(x); since the latter splits in Fpn, so does the former, and the set of roots of fm(x) forms a subfield of Fpn of size pm). W 4/12 33 Examples of extending embeddings: three extensions of Q → C to Q[x]/⟨x3 – 2⟩, all with different images. Six extensions of Q → C to Q[x, y]/⟨x3 – 2, y2 + y + 1⟩, all with the same image Q(21/3, ω) ⊂ C. Proof of parts of the embedding extension theorem. Given a field extension L over K and an embedding σ of K to an auxiliary field F, there are (1) no more than [L : K] extensions of σ to L. More precisely, (2) up to replacing F with a finite extension, (a) there is always at least one extension of σ to L, and (b) there are exactly [L : K] extensions of σ to L if and only if L/K is separable. (See Theorem 3.8 in K. Conrad’s notes on separability.) M 4/10 32 Perfect fields (no inseparable algebraic extensions), and connection to surjectivity of the Frobeius endomorphism (BB 8.2) in characteristic p. Statement of the embedding extension theorem: Starting with a field embedding σ : K → F, we aim to extend σ it to a finite extension L of K. Theorem: there are at most [L : K] extensions. More precisely, up to replacing F with a big enough finite extension, (a) there is always at least one way of extending σ to L; (b) there are exactly [L : K] ways of extending σ if and only if L/K is separable; (c) all the ways of extending σ have the same image in F if and only if L/K is a splitting field. (BB Lemma 8.1.5, Theorem 3.8 in K. Conrad’s notes on separability, plus epsilon.) F 4/7 31 Separability! Formal derivative operator for polynomials. Bilinear maps and a proof of the product rule. A polynomial has no multiple roots in any extension field if and only if it is relatively prime to its derivative. Separability for (irreducible) polynomials, elements and extensions. If irreducible f in K[x] is inseparable, then K has prime characteristic p, and f(x) = g(xp) for some (irreducible) g in K[x]. W 4/5 30 Consequences of splitting fields for finite fields. A finite field F is a finite extension of Fp for some prime p, so has pn elements for n = [F : Fp]. Since F× is a cyclic group, the extension F/Fp is automatically simple: any multiplicative generator of F× will be certainly be a field generator for F. Every element of F× has multiplicative order dividing pn – 1, so that the pn elements of F are all the roots of xpn– x. In other words, F is a splitting field for xpn– x. Since splitting fields are unique up to isomorphism, any two fields of pn elements are isomorphic. So up to isomorphism, there is at most one finite field of each prime power order. M 4/3 29 Splitting fields. Examples. Existence theorem. Uniqueness (up to isomorphism) theorem. See Theorem 6.4.2 and Theorem 6.4.5. F 3/31 28 Examples of automorphism groups for finite simple extensions: Q(i)/Q, Q(∛2)/Q, Q(ω, ∛2)/Q(ω), Q(√2 + √3)/Q, F3(t1/3)/F3(t). W 3/29 27 A bit about distinguished classes of field extensions (closed under towers and translation). Algebraic extensions of algebraic extensions are algebraic. For a finite simple extension L = K(α) of K, the set of automorphisms in Aut(L/K) is in bijection with K-linear field isomorphisms K[x]/⟨m(x)⟩ ≅ L, where m(x) is the minimal polynomial of α over K. These in turn are in bijection with distinct roots of m(x) in L. In particular, #Aut(L/K) ≤ [L:K], with equality if and only if m(x) splits completely into distinct linear factors in L. See also section VII.1.2, especially Corollary 1.7, of Aluffi, Algebra: Chapter 0. M 3/27 26 An element over a field is algebraic if and only if it is contained in a finite extension. Degree of an element divides degree of any field extension it is contained in. An extension generated by finitely many algebraic elements is finite, of degree at most the product of the degrees of the generators. Finite simple extensions stay finite under translation. Example: Q(21/2, 21/3)/Q and Q(51/2, 21/3)/Q. Realizing each as a simple extension. Set of algebraic elements of an extension forms a field. Algebraic extensions. Example of an infinite algebraic extension: Q, the subset of elements of C that are algebraic over Q. F 3/24 25 Degree of a field extension. Finite and infinite field extensions. Degrees multiply in towers (Theorem 6.2.5). Degree of quotient ring K[x]/⟨f(x)⟩ over a field K is the degree of f(x). Degree of an element. Yet more group work on simple extensions. W 3/22 24 No class. One-on-one meetings for midterm mathematical conversations. M 3/20 23 Review of vector spaces: definition, subspaces, spanning sets, linearly independent sets, bases. Theorem: Every vector space that has a finite spanning set has a basis, and any two bases have the same cardinality, which we call the dimension of the vector space. (The theorem is true without the finite spanning set requirement, but the proof requires Zorn’s lemma.) If L over K is a field extension, then L is a vector space over K. More group work on simple extensions. F 3/17 22 Simple extensions: Suppose L = K(α) is a simple field extension of a field K. Then either α is algebraic over K, in which case the kernel of the evaluation-at-α homomorphism is a nonzero principal ideal generated by the minimal polynomial f(x) of α, which is irreducible, and L = K(α) = K[α] ≅ K[x]/⟨f(x)⟩. Or α is transcendental over K, in which case the evaluation-at-α homomorphism is injective and L = K(α) ≅ K(x), where K(x) = Q(K[x]) is the field of rational functions over K. Group work on simple extensions. W 3/15 21 Field extensions. Examples and nonexamples. Algebraic and transcendental elements of a field extension. Examples. Minimal polynomial of an algebraic element of a field extension. Simple extensions. M 3/13 20 Quotient fields are unique (because they satisfy the universal mapping property for quotient fields!). Characteristic of a ring. The characteristic of a domain is either zero or prime. Every field contains a copy either of Q or of Zp for some prime p. F 3/3 19 In a PID every nonzero prime ideal is maximal. Notation (a, b) for gcd(a, b) vs. ⟨a, b⟩ for the ideal generated by a and b. Associate elements of a ring. Quotient fields/fields of fraction: the construction (see Theorem 5.4.4) and the universal mapping property (Thereom 5.4.6). W 3/1 18 Prime and maximal ideals. Examples in Z, field F, F[x], Z[x]. Ideal J of R is prime if and only if R/J is an integral domain, maximal if and only if R/J is a field. Maximal ideals are prime. M 2/27 17 First isomorphism theorem for rings (or as BB calls it “fundamental homomorphism theorem for rings”): read proof of Theorem 5.2.6. Group work: computing quotient rings: Z/3Z, Q[x]/⟨x2 − 2⟩, Q[x]/⟨x2 + x + 1⟩, Z[x]/⟨x2 + 1⟩, Z[x]/⟨x⟩, Z[x]/⟨5⟩, Z[x]/⟨2, x⟩, Z[x]/⟨2x⟩. Correspondence between ideals of R/J and ideals of R containing J: read proof of Proposition 5.3.7. F 2/24 16 Quotient rings: ring structure on R/J, universal mapping property of quotient rings. Group work on blackboard: first isomorphism theorem for rings. W 2/22 15 Trivial ideal and unit ideal. Intersection of ideals; sum of ideals. Ideal generated by a set of elements. A nonzero ring is a field if and only if it has exactly two ideals. Prinicipal ideal domains: Z and F[x] are PIDs, Z[x] is not a PID. T 2/21 14 Every ring receives unique map from Z and sends unique map to the zero ring. Review: first isomorphism theorem for abelian groups. Example: exp : R → C×. Evaluation homomorphism. Example: evaluation-at-√2 Q[x] → R. Quiz feedback. F 2/17 13 HW #2 return: problems 1, 3, 4, 6, 7, 10, 11, 14 graded. Product of two rings: when a domain, units. The ring isomorphism Zmn ≅ Zm × Zm if gcd(m, n) = 1, and consequences for the Euler ϕ-function (it’s multiplicative). The polynomial ring R[x]: domain if R is, in which case R[x]× = R×. Quiz W 2/15 12 Well-definition of functions whose domain is naturally a set of equivalence classes (such as Zm or Q). Finite integral domains are fields. Subrings. Ring homomorphisms. Images are subrings. Ideals: definitions. HW #1 return: problems 1, 3, 4, 6, 7, 9, 10 graded. M 2/13 11 BB 5.1: Rings, units, zero divisors, integral domains; definitions and examples. F 2/10 W 2/8 M 2/6 10 9 8 Prof. Pollack: BB 4.3 and 4.4 F 2/3 7 Prof. Pollack: BB through end of 4.2. W 2/1 6 Prof. Weinstein: Parts of 4.3 in BB and finite fields of size ≤ 8. M 1/30 5 Prof. Pollack. BB through Theorem 4.2.4. F 1/27 4 BB to end of 4.1. Also: group of units of a (commutative) ring. W 1/25 3 BB 4.1 through Proposition 4.1.5 M 1/23 2 Symmetries of Q[√2], Q[∛2], and Q[∛2, ω]. F 1/20 1 Syllabus. Symmetries of Q[√2] preserving addition and multiplication. Older announcements Zorn’s lemma is a proposition of set theory. It is surprisingly equivalent to the axiom of choice, which is much more intuitive: it says that the product of arbitarily many nonempty sets is nonempty. One needs it to prove that every vector space has a basis, that every ring has a maximal ideal, and so on. Some of these applications appear as optional problems on HW #5; a number of applications are worked out in Keith Conrad’s blurbs (see below). If you’re just looking to understand what Zorn’s lemma says, with lots of examples, start by reading the introduction to Keith Conrad’s first blurb on Zorn’s lemma. There’s also a part II. Another writeup that may be helpful: Tim Gowers on how to use Zorn’s lemma. Why don’t we allow the zero ring to be called a field? “Too simple to be simple” on nLab Pete Clark’s answer to a question on math.stackexchange. Why do we assume that every ring has a multiplicative identity? Bjorn Poonen: “Why all rings should have a 1” For an example of two associates that are not unit multiples of each other, see the aptly titled “When are associates unit multiples?,” Rocky Mountain Journal of Mathematics, 2004. Do let me know if you discover a simpler example! German mathematicial Richard Dedekind lived 1831–1916, developed the Dedekind cut construction of the reals starting in 1858, and the theory of ideals (in the context of unique factorization in “rings of integers” of finite extensions of Q) starting in the 1870s. Resources Supplementary textbooks Judson’s Abstract algebra is a free online algebra textbook that roughly covers the same material as Beachy and Blair. Fraleigh’s A first course in abstract algebra (7th and 8th editions both fine; earlier ones may be as well) is another option, long and lots of examples. Keith Conrad at UConn has written many excellent short blurbs on various topics in undergraduate algebra. One that might be helpful this week: Well-defined functions Tutoring room: TBA