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 - normality

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 \(p^r\) 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 - separability

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

Book: 8.14, 8.15, 9.18

For 8.14, use Gauss’s lemma to relate \(F[x,y]\) and \(F(x)[y]\) in order to show that \(y\) divides \(f(x+y)-f(x)\) in \(F[x,y]\). Do not write out any sort of expansion of \(f(x+y)\); the point of this formulation is to avoid that.

Additional:

  1. Show that every element of the finite field \(\mathbb F_{p^r}\) has a unique \(p\)th root in \(\mathbb F_{p^r}\).

  2. Let \(k\) be a field of characteristic \(p\), and suppose \(\alpha\) is separable over \(k\). Show that \(k(\alpha) = k(\alpha^{p^n})\). Is the converse true? Try to prove it.

  3. Let \(k\) be a field of characteristic \(p\) and \(u,v\) transcendental over \(k\). Prove that \(k(\sqrt[p] u,\sqrt[p]v)\) has degree \(p^2\) over \(k(u,v)\). Bonus: show that there are infinitely many sub-extensions of \(k(u,v)\) in \(k(\sqrt[p] u,\sqrt[p]v)\).

HW7 - Galois Theory

Due: Friday, April 3rd, at 10pm. See Blackboard for the gradescope link.

Book: 9.22, 9.23, 9.25, 9.31, 9.32.

Additional:

  1. Calculate the splitting field of \(x^4 - 2\) over \(\mathbb Q\). Determine its Galois group and all of its subgroups. Use these to write down all of its subfields (in the form \(\mathbb Q(\alpha)\)). Draw diagrams of both the subgroups and subfields.

  2. Let \(K\) be a finite extension of the finite field \(\mathbb F_p\). Use the group theory lemma on cyclic groups to prove that \(\Gal(K/\mathbb F_p)\) is cyclic without using the Frobenius (as we did in class).

HW8 - cyclotomic, radical, and solvable extensions

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

HW9 -

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

HW10 -

Due: Friday, April 24th, at 10pm. See Blackboard for the gradescope link.

HW11 -

Due: Friday, May 1st, at 10pm. See Blackboard for the gradescope link.

HW12 -

Due: Friday, May 8th, at 10pm. See Blackboard for the gradescope link.