Homework

See below for the assignments and their due dates.

HW1 - review

Due: Friday, January 30th, at 10pm. See Blackboard for the gradescope link.

Book: 3.2, 3.4, 3.5, 3.6, 3.7.

Additional: calculate the following

  1. \(\mathbb Q[x]/(x^2 + 5)\),
  2. \(\mathbb Q[x]/(x^2 + 1)\),
  3. \(\mathbb Q[x]/(x^2 - 1)\).

Comments:

  • Book 3.5: very important.
  • Book 3.7: an example related to a question from after class Thursday. We distinguish formal polynomials over \(R\), which comprise \(R[x]\), and polynomials as functions from \(R\) to \(R\). Remember that the evaluation map allowed for evaluation at things outside the ring \(R\) – we will soon (Chap 5) have some good concrete examples to make this point.
  • Calculations: think about what makes (c) so different. Also, you can (should) use long division, which we will develop a bit more rigorously later on.

HW2 - review, starting fields

Due: Friday, February 6th, at 10pm. See Blackboard for the gradescope link.

Book: 3.41, 3.43, 3.51, 3.52, 3.53, 5.3, 5.6.

Additional:

  1. Recall \(\mathbb R[x]/(x^2 + 1) \cong \mathbb C\). Now calculate \(\mathbb C[x]/(x^2+1)\) (cf last homework (c)).

  2. Determine, with proof, which of the following ideals of \(\mathbb Z[i]\) are prime:

    1. \((2)\)
    2. \((3)\)
    3. \((1+i)\)
    4. \((10)\)
    5. \((2+i)\)
    6. \((5)\)
    7. \((3+i)\)

Comments:

  • Book 3.51: this ideal is usually written \((2,x)\)
  • Book 3.53: \(R^*\) is the set of \(r\) in \(R\) which have a multiplicative inverse, meaning that there is some \(s\) in \(R\) such that \(rs = 1\). It’s the largest subset of \(R\) which forms a group under multiplication.

HW3 - more on fields

Due: Friday, February 13th, at 10pm. See Blackboard for the gradescope link.

Book: 5.1, 5.7, 5.9, 5.10, 5.19, 5.20, 5.21, 5.24a,b

Additional:

  1. Observe that \[\mathbb F_p [x]/(x^2+1) \cong \mathbb Z[x]/(p,x^2+1) \cong \mathbb Z[i]/(p).\] With (a) in mind, use this to characterize the primes \(p\) in \(\mathbb Z\) which remain prime in \(\mathbb Z[i]\): these are precisely those primes congruent to \(3\) mod \(4\). You will need to use the fact that \(\mathbb F_p^*\) is cyclic. This is a small piece of “quadratic reciprocity”, which we may prove later.

  2. The ring \(\mathbb Z[\rho]\) where \[\rho = \frac{1 + \sqrt{-3}}{2}\] is a PID. Recall that \(\rho^3 = 1\) and \(\rho \neq 1\). Using an approach similar to (a), classify the primes \(p\) such that \(-3\) is a square mod \(p\). Hint: you’ll still want to use the the complex norm given by \(\alpha\bar\alpha\), which you will need to prove is an integer. This is the special case of quadratic reciprocity for \(-3\).

Comments:

  • Book 5.10: ignore the hint, there is an easier one-line approach.

HW4 - irreducibility and Gauss’s lemma

Due: Friday, February 20th, at 10pm. See Blackboard for the gradescope link.

Book: 8.1, 8.2, 8.5, 8.8, 8.9, 8.18, 8.19, 8.20

HW5 - splitting fields

Due: Friday, February 27th, at 10pm. See Blackboard for the gradescope link.

Book: 9.3 – 9.13

Additional:

  1. For each of the following fields \(K\) and polynomials \(f\) in \(K[x]\), calculate the splitting field of \(f\) over \(K\) and the degree of the splitting field over \(K\).

    1. \(K = \mathbb Q\) and \(f(x) = x^7 - 6\),
    2. \(K = \mathbb Q(\zeta_{10})\) and \(f(x) = x^5 - 13\),
    3. \(K = \mathbb Q(\zeta + \zeta^{-1})\) and \(f(x) = x^{13} - 2\).

  2. Suppose that \(K(\alpha)\) has odd degree over \(K\). Prove that \(K(\alpha^2) = K(\alpha)\).

  3. (over \(\mathbb Q\)) Prove that \(\Phi_{p^r}(x) = \Phi_{p^{r-1}}(x^p)\) when \(r > 1\) and use this to prove directly that \(\Phi_{p^r}(x)\) is irreducible with Eisenstein’s criterion.

  4. Consider the polynomial \(f(x) = x^{p^r} - x\) over \(\mathbb F_p\). Prove that:

    1. \(f(x)\) has \(p^r\) distinct roots,
    2. the roots of \(f(x)\) form a field containing \(\mathbb F_p\),
    3. the splitting field of \(f(x)\) is a degree \(p\) extension of \(\mathbb F_p\)

Therefore, there is an extension of \(\mathbb F_p\) of degree \(d\) for every positive integer \(d\), and this extension is unique (within a given algebraic closure).

  1. Consider a finite field \(\mathbb F_p\). Let \(q^s\) be a power of some prime. How many monic irreducible polynomials are there in \(\mathbb F_p[x]\) of degree \(q^s\)?

HW6 - Galois groups

Due: Friday, March 17th, at 10pm. See Blackboard for the gradescope link.