MAT 424 Assignments

Highlighted assignments constitute recent material.
Keep in mind that even if a problem isn't assigned, you should still read it, as we may well use it later.
 

Topic

 Assignments
Also on 12 May 2016 (for reasons given in email)
  • Questions 9⋅54, 9⋅55, 9⋅64 (similar to the proof in the notes of Lemma 9⋅57)
  • Is the center of a group (\(Z\) in class, \(Z(G)\) in the notes) a normal subgroup? Either prove it is or provide a counterexample.
  • Is the centralizer of an element (\(C_g\) in class, \(C_g(G)\) in the notes) a normal subgroup? Either prove it is or provide a counterexample.
  • Let \(G\) be a finite group with \(n=|G|\). If \(m>0\) and \(m\mid n\), does \(G\) necessarily have a subgroup \(H\) such that \(|H|=m\)? (That is, does Cauchy’s Theorem generalize?)
  • Compute the class equation for \(Q_8\); that is, determine \(|Z|\) and \(|C_g|\) for each \(g\) in a suitable partition of \(G\setminus Z\).
  • Let \(f(x)=x^5-1\). Compute the splitting field \(\mathbb E\) of \(f\) over \(\mathbb Q\) (that is, the smallest field extension where \(f\) factors into linear polynomials). Determine \([\mathbb E:\mathbb Q]\) and \(\mathrm{Gal}(\mathbb E/\mathbb Q)\).
  • Let \(f(x)=x^5-2\). Compute the splitting field \(\mathbb E\) of \(f\) over \(\mathbb Q\) and determine \([\mathbb E:\mathbb Q]\).
12 May 2016 Questions 9⋅37, 9⋅38, 9⋅40, 9⋅44, 9⋅50, 9⋅51, 4⋅151(c,d)
3 May 2016 Questions 9⋅24, 9⋅28 (mostly done in class, so finish it), 9⋅29, 9⋅30, 9⋅33
26 Apr 2016 Questions 9⋅2 (“Show that” really should be “Explain why”), 9⋅4, 9⋅5, 9⋅7 (factor it over \(\mathbb Q\) first), 9⋅9, 9⋅12, 9⋅20, 9⋅21(a,d)
14 Apr 2016 Questions 7⋅22, 7⋅27, 8⋅18, 8⋅19, 8⋅20, 8⋅21, 8⋅24, 8⋅25
12 Apr 2016 Read Sections 8⋅2 and 8⋅3.
7 Apr 2016
  • Questions 6⋅99, 6⋅100, 6⋅103, 6⋅105
  • Let \(f(x) = x^8 + 5x^7 + 9x^5 + 53x^4 + 40x^3 + 72x + 360\). We want to factor \(f\) over \(\mathbb Z\) by first factoring over \(\mathbb Z_p\) for some “good” values of \(p\).
    • Suppose we factor \(f\) over \(\mathbb Z_3\). Someone might argue that this is actually a bad idea, because it gives false positives; that is, it allows too much factorization. Why?
      Hint: Think about an “obvious” factorization of \(f\) when you write its coefficients modulo 3, and whether this “obvious” factorization is also true over \(\mathbb Z\).
    • Based on the answer above, what would be bad moduli for factoring \(f\)?
    • If we wanted to factor \(f\) over \(\mathbb Z_p\) for several irreducibles \(p\), then reconstruct the factorization over \(\mathbb Z\) using the Chinese Remainder Theorem, without using for \(p\) any of the moduli you identified in your previous answer, and you wanted to use the smallest \(p\) possible, how many such \(p\) would you want to use, and what are they?
    (I realize I didn’t talk much about this latter question; I’m not expecting “the correct” answer so much as “an intelligent” answer, based on what you are supposed to know. I do not expect you to factor \(f\). That said, if you’re rather bored and want to try using the Distinct Degree and Equal Degree factorization algorithms, I can tell you that it has three binomial factors of different degrees, and it could well take you a very long time to factor by hand.)
5 Apr 2016 Read Section 7⋅2.
29 Mar 2016 Questions 6⋅62, 6⋅65, 6⋅67, 6⋅71, 6⋅73, 6⋅79, 6⋅80, 6⋅92, 6⋅93
10 Mar 2016 Questions 6⋅6, 6⋅9, 6⋅11, 6⋅12 (not 2, which was done in class), 6⋅13, 6⋅14, 6⋅19, 6⋅22, 6⋅23, 6⋅28, 6⋅32, 6⋅42, 6⋅43, 6⋅48, 6⋅50, 6⋅58, 6⋅59
3 Mar 2016 Read Section 6⋅3.
25 Feb 2016
  • Read Sections 6⋅1, 6⋅2
  • 5⋅46, 5⋅54, 5⋅56, 5⋅57, the oddly-numbered 5⋅3 at the end of the chapter
18 Feb 2016
  • Read Sections 5⋅5, 5⋅6
  • Questions 5⋅31, 5⋅32, 5⋅33, 5⋅42
  • In Question 5⋅42, you built a Cayley table for \(\mathbb Z_{15}^*\). This group under multiplication has the same number of elements as \(\mathbb Z_n\) under addition for a certain value of \(n\). Which value of \(n\)? Are they actually isomorphic? If so, describe an isomorphism (i.e., state which element of \(\mathbb Z_{15}^*\) maps to which element of \(\mathbb Z_n\)). If not, explain why not.
11 Feb 2016
  • Read Sections 5⋅3, 5⋅4
  • Questions 5⋅10, 5⋅11, 5⋅16, 5⋅18, 5⋅20, 5⋅21, 5⋅22, 5⋅30
  • Show that \(\gcd(n,n-1)=1\) for any integer \(n\). The argument when \(n=0\) might be a little different.
  • Let \(R=\mathbb C[x]\) and \(A=\left< x^{12}-2x^6+1\right>\).
    • What are the roots of \(A\)’s generator? (List all six!)
    • Are those roots also roots of the other polynomials of \(A\)? Why or why not?
    • It turns out that \(\sqrt A\) is a principal ideal. What polynomial generates \(\sqrt A\)?
    • List an element of \(\sqrt A\) that is neither an element of \(A\) nor the generator of \(\sqrt A\).
    • Are the roots of \(A\)’s generator also roots of \(\sqrt A\)’s generator? Why or why not?
2 Feb 2016 Consider \(R=\mathbb Z_{15}\) as a ring under modular addition and multiplication.
  1. List the elements of \(\left<2\right>\) in \(R\).
  2. List the elements of \(\left<3\right>\) in \(R\).
  3. List the elements of \(\left<5\right>\) in \(R\).
  4. Why aren't the elements of \(\left<0\right>\) or \(\left<1\right>\) interesting?
  5. Of the ideals listed above, list all that are maximal. Be sure to explain why they are maximal; i.e., show that adding just one more element gives you the entire ring.
  6. For one of these maximal ideals, list its cosets, and build the Cayley table.
  7. Is the quotient ring you just considered a field?
  8. Bonus: Compare the results from class with the ones here. Do you notice any pattern to the generators of the ideals that equal \(R\)?
  9. Awesome Bonus: Prove or disprove that this pattern holds in general.
28 Jan 2016 Let \(\mathbb F\) be any field, and \(R=\mathbb F[x,y]\). Let \(A\) be the ideal \(\left< xy,xy+2\right>\).
  1. List at least five elements of \(A\). For full credit, find at least one element that comes from cancellation.
  2. Explain how we know \(A=R\). Note: This is a little harder than the one in class, but not by much. Remember that the polynomials are over a field and so elements of the field itself are also polynomials (they are considered constant polynomials).
  3. Explain how we know the polynomials of \(A\) have no common roots.
26 Jan 2016 Recall that \(\Omega_n\) is the group of \(n\)th roots of unity.
  1. List the elements of \(\Omega_3\). (Do not just write \(\{\omega,\omega^2,\ldots\}\) — I want numbers in the form \(a+bi\).)
  2. List the elements of \(\Omega_{12}\). (Do not just write \(\{\omega,\omega^2,\ldots\}\) — I want numbers in the form \(a+bi\).)
  3. Sketch the elements of both \(\Omega_3\) and \(\Omega_{12}\) on the complex plane. Label each element clearly.
  4. How do we know \(\Omega_3 < \Omega_{12}\)?
  5. How do we know \(\Omega_3 \triangleleft \Omega_{12}\) without checking equality of cosets?
  6. What is the size of \(\Omega_{12} / \Omega_3\)?
  7. How do we know \(\Omega_{12} / \Omega_3\) is a group, without checking all four properties?
  8. How do we know \(\Omega_{12} / \Omega_3\) is an abelian group, without checking every product?
  9. Build a Cayley table of \(\Omega_{12} / \Omega_3\).
  10. Build a homomorphism from \(\Omega_{12}\) onto \(\mathbb{Z}_4\) that has \(\Omega_3\) as its kernel. Don’t forget to show that it’s a homomorphism. What does the Isomorphism Theorem tell us about these three groups?
21 Jan 2016 Show that the subgroup \(\{\iota,\rho,\rho^2\}\) of \(D_3\) is normal, but the subgroup \(\{\iota,\varphi\}\) is not.