Course webpage for MA 541, Modern Algebra II, Spring 2023

Instructor: Anna Medvedovsky


  • 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).

  • Starting-off questions for our midterm mathematical conversations

  • Quiz

Older announcements
Homework assignments

Post date Due date HW assignment
Final 3/25/23

HW #5

Edited 3/5/23

HW #4

Final 2/17/23
2/27 also ok

HW #3

Final 2/3/23
2/13 in class

HW #2

Final 1/23/23

HW #1

Class meetings

Date Class Topics
M 3/27
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
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
No class. One-on-one meetings for midterm mathematical conversations.
M 3/20
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
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
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
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
In a PID every nonzero prime ideal is maximal. Notation (a, b) for gcd(ab) vs. ⟨ab⟩ 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
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
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
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
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
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
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×.


W 2/15
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
BB 5.1: Rings, units, zero divisors, integral domains; definitions and examples.
F 2/10
W 2/8
M 2/6
Prof. Pollack: BB 4.3 and 4.4
F 2/3
Prof. Pollack: BB through end of 4.2.
W 2/1
Prof. Weinstein: Parts of 4.3 in BB and finite fields of size ≤ 8.
M 1/30
Prof. Pollack. BB through Theorem 4.2.4.
F 1/27
BB to end of 4.1. Also: group of units of a (commutative) ring.
W 1/25
BB 4.1 through Proposition 4.1.5
M 1/23
Symmetries of Q[√2], Q[∛2], and Q[∛2, ω].
F 1/20
Syllabus. Symmetries of Q[√2] preserving addition and multiplication.

Older announcements
  • Why don’t we allow the zero ring to be called a field?
  • Why do we assume that every ring has a multiplicative identity?

  • 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.

  • 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.

  • Tutoring room: TBA