| Days | Time | Location | CRN |
|---|---|---|---|
| Mon Wed Fri | 14:00-14:50 | Benedum Hall 227 | 18944 |
My office is Thackeray 518. I will hold office hours Monday 4-5 and Thursday 3-4 and am also available by appointment.
I will be holding extra office hours during finals week: 5-8pm on Sunday, April 24 in Thackeray 525, 4-5pm on Monday, April 25 in Thackeray 518 and 5-8pm on Wednesday, April 27 in Thackeray 525.
The midterm was on March 2 and the final will be on April 30. These are the exams from Math 430 last spring.
| Practice Midterm | |
| Practice Final | Practice Final Solutions |
| 2016 Midterm Solutions |
The midterm covered Chapters 1,2,3,4,5,6,9,10 from the textbook, excluding Section 4.3. The final will be cummulative, also covering Chapters 11, 16, 17, 18, as well as Theorem 13.5 and the Fundamental Theorem of Algebra.
| Due | Problems | Solutions | LaTeX Solutions |
|---|---|---|---|
| Jan 22 | Problem Set 1 | Solutions | LaTeX |
| Feb 5 | Problem Set 2 | Solutions | LaTeX |
| Feb 19 | Problem Set 3 | Solutions | LaTeX |
| Mar 18 | Problem Set 4 | Solutions | LaTeX |
| Apr 1 | Problem Set 5 | Solutions | LaTeX |
| Apr 18 | Problem Set 6 | Solutions | LaTeX |
| Apr 22 | Problem Set 7 | Solutions | LaTeX |
| Due | Problems |
|---|---|
| Apr 30 | Sage Problem Sets |
| Due | Problem | Remarks |
|---|---|---|
| Jan 8 | Find a set with no proper subsets. | \(\{\varnothing\}\) is the set containing the empty set, not the empty set itself. |
| A one element set has \(\varnothing\) as a proper subset. | ||
| Jan 11 | Find two functions whose composition is injective, but they are not both injective. | When defining a function, its good practice to specify the domain and codomain. |
| Note that \(\mathbb{N} \to \mathbb{Z}, x \mapsto \sqrt{x}\) is not a function (the square root of \(2\) is not an integer for example). | ||
| When defining a function, either write \(f(x) = \sqrt{x}\) or \(f : x \mapsto \sqrt{x}\). \(\{\sqrt{x} \vert x \in A\}\) is not correct. | ||
| Note that function composition is not the same as multiplication! The composition of \(x \mapsto x^3\) with itself is \(x \mapsto x^9\), not \(x \mapsto x^6\). | ||
| Order matters in composition! If you first square the input and then take its square root you get the absolute value map from \(\mathbb{R}\) to \(\mathbb{R}^+\), which is not injective. If you first take the square root and then the square, you get the inclusion of \(\mathbb{R}^+\) into \(\mathbb{R}\), which is. | ||
| The question was about \(f\) and \(g\) being injective, not about finding an example where one of \(f\circ g\) and \(g\circ f\) was injective while the other wasn't. | ||
| Jan 13 | Find a relation on \(\mathbb{Z}\) (the integers) that is transitive, NOT symmetric and NOT reflexive. | |
| Jan 15 | Find the partitions of the set \(\{1,2,3,4,5\}\). | |
| Jan 20 | Find two integers, both larger than a billion, that are relatively prime. | |
| Jan 22 | A primitive element modulo \(n\) is a number \(a\) so that the powers of \(a\) modulo \(n\) cover all of the conjugacy classes that are relatively prime to \(n\). Find an \(n\) where a primitive element does not exist. | |
| Jan 25 | No Daily Homework. | |
| Jan 27 | Find all of the subgroups of \(D_4\). | |
| Jan 29 | Find the smallest subgroup of \(\mathbb{Z}_{32}\) containing both \(12\) and \(18\). | |
| Feb 1 | Find all of the cyclic subgroups of \(U(30) = \mathbb{Z}_{30}^\times\). | |
| Feb 3 | Find a primitive \(12\)th root of unity (in the form \(a + bi\)). | |
| Feb 5 | Find the order of the permutation \((1)(2,3)(4,5,6)(7,8,9,10)(11,12,13,14,15)(16,17,18,19,20,21)\). | |
| Feb 8 | Find the subgroups of \(A_4\). | |
| Feb 10 | Find the center of \(D_4\). | |
| Feb 12 | No Daily Homework. | |
| Feb 15 | Find the left and right cosets of the subgroup generated by \((123)\) within \(A_4\). | |
| Feb 17 | No Daily Homework. | |
| Feb 19 | Find an isomorphism between \(\mathbb{Z}_6\) and \(U(18) = \mathbb{Z}_{18}^\times\). | |
| Feb 22 | Find an isomorphism between \(\mathbb{Z}_{10}\) and \(\mathbb{Z}_2 \times \mathbb{Z}_5\). | |
| Feb 24 | Find an isomorphism between \(\mathbb{Z}_{12} \times \mathbb{Z}_{18}\) and \(\mathbb{Z}_{36} \times \mathbb{Z}_6\). | |
| Feb 26 | No Daily Homework. | |
| Feb 29 | For each subgroup \(H\) of \(Q_8\), determine whether or not it's normal, and if it is, identify the quotient \(Q_8/H\) from the list \(\{1\}, \mathbb{Z}_2, \mathbb{Z}_4, \mathbb{Z}_2 \times \mathbb{Z}_2, \mathbb{Z}_8, \mathbb{Z}_4 \times \mathbb{Z}_2, \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2, D_8, Q_8\) (up to isomorphism). | |
| Mar 2 | No Daily Homework. | |
| Mar 4 | No Daily Homework. | |
| Mar 14 | No Daily Homework. | |
| Mar 16 | Find all homomorphisms \(Q_8 \to \mathbb{Z}_4\). | |
| Mar 18 | List the abelian groups of order \(72\) up to isomorphism. | |
| Mar 21 | Find two integral domains \(R\) and \(S\) and a zero divisor in \(R \times S\). | |
| Mar 23 | Find an infinite integral domain with characteristic 5. | |
| Mar 25 | Find a group homomorphism \(\mathbb{Z}_5 \to \mathbb{Z}_5\) which is not a ring homomorphism. | |
| Mar 28 | No Daily Homework. | |
| Mar 30 | Find an ideal in the polynomial ring \(\mathbb{Q}[x]\) that is prime but not maximal. | |
| Apr 1 | Find a solution to the system of equations \(x \equiv 3 \pmod{5}\) and \(x \equiv 1 \pmod{7}\) and \(x \equiv 2 \pmod{8}\). | |
| Apr 4 | No Daily Homework. | |
| Apr 6 | Find the greatest common divisor of \(x^4 - 2x^3 + 2x^2 - 2x + 1\) and \(4x^3 - 6x^2 + 4x - 2\). | |
| Apr 8 | No Daily Homework. | |
| Apr 11 | Find an irreducible, monic polynomial in \(\mathbb{Z}[x]\) of degree 37 with coefficient of \(x^5\) equal to \(-8\). | |
| Apr 13 | Find two distinct factorizations of \(6 \in \mathbb{Z}[\sqrt{-5}]\). | |
| Apr 15 | No Daily Homework. |
Here are some resources for learning and improving your LaTeX skills.
Sage is an open source mathematics software package that is capable of computing with many of the objects that we will be learning about this semester. Moreover, there is a version of our textbook that includes sections on using Sage and Sage exercises.
If you are interested in learning about Sage, I will make (optional) extra credit homework assignments available. Here are some resources (beyond what is in the textbook) for learning about Sage.