2025 Algebra 2

Homework 1 :: Due on Friday 28 Feb 2024 by 11:59pm on Gradescope.

1. Exercise 12.1.5.
2. Exercise 12.2.4.
3. Exercise 12.2.9.

Spend your time on 1–6. Do 7 and 8 only if you have time.

1. True or false?
   1. The element \(x\) is a prime element of \(\mathbf{Q}[x,y]\).
   2. The element \(x\) is an irreducible element of \(\mathbf{Q}[x,y]\).
   3. The element \(2x+1\) is a prime element of \(\mathbf{Z}[x]\).
   4. The element \(2x+1\) is an irreducible element of \(\mathbf{Z}[x]\).
2. Recall the definitions of ring homomorphism, kernel, image, and the first isomorphism theorem.
3. Let \(R\) be a ring and \(x\) a variable. Recall the definition of the polynomial ring \(R[x]\), or more generally, the polynomial ring \(R[x_1, \dots, x_n]\).
4. Let \(R = \mathbf{Z}/p \mathbf{Z}\). Show that the polynomial \(x^p-x\) evaluates to zero for all \(x \in R\). Is \(x^p-x = 0\) in \(R[x]\)?

5. Convince yourself that a ring homomorphism \(R[x] \to S\) is the same thing as a ring homomorphism \(R \to S\) and an element of \(S\) that serves as the image of \(x\).

   That is, given a ring homomorphism \(\psi \colon R \to S\) and an element \(s \in S\), there is a unique ring homomorphism \(R[x] \to S\) which agrees with \(\psi\) on degree zero polynomials and sends \(x\) to \(s\).

6. Consider \(z - xy \in \mathbf{C}[x,y,z]\). Show that it is irreducible in two ways:
   1. identify the quotient \(\mathbf{C}[x,y,z]/(z-xy)\) and conclude that \(z-xy\) is prime.
   2. Show that it is primitive as an element of \(\mathbf{C}[x,y] [z]\) and irreducible in \(\mathbf{C}(x,y)[z]\).
7. Consider the ring homomorphism \(\phi \colon \mathbf{Z}[x] \to \mathbf{Q}\) that sends \(x\) to \(1/2\). What is the kernel of \(\phi\)?
8. Consider the ring homomorphism \(\phi \colon \mathbf{Q}[x] \to \mathbf{R}\) that sends \(x\) to \(1 + \sqrt 2\). What is the kernel of \(\phi\)?

Homework 2 :: Due on Friday 7 Mar 2025 by 11:59pm on Gradescope.

1. Exercise 12.3.4.
2. Exercise 12.4.4.

Homework 3 :: Due on Friday 14 Mar 2025 by 11:59pm on Gradescope.

1. Exercise 12.M.4 (a) and (b). In (a), please describe an explicit isomorphism \(\mathbf{R}[x,y]/(x^2+y^2-1) \to R\). (Caution: \(\mathbf{R}[x,y]\) is not a principal ideal domain.)
2. Exercise 15.2.3.

1. How would you prove that \(\sqrt 3 \not \in \mathbf{Q}[\sqrt 2]\)? The most simple minded way is to prove that for no \(a, b \in \mathbf{Q}\), we have \((a+b\sqrt 2)^2 = 3\). Here is another way that uses the concept of conjugates. We say that \(\alpha, \beta \in \mathbf{C}\) are conjugates over \(\mathbf{Q}\) if for every \(p(x) \in \mathbf{Q}[x]\), if \(p(\alpha) = 0\) then \(p(\beta) = 0\) and vice versa. Equivalently, \(\alpha\) and \(\beta\) are conjugates over \(\mathbf{Q}\) if they have the same minimal polynomial over \(\mathbf{Q}\).
   1. Prove that \(a+b \sqrt 2\) and \(a-b \sqrt 2\) are conjugates over \(\mathbf{Q}\).
   2. Suppose \(\sqrt 3 = a + b \sqrt 2\). Take \(p(x) = x^2-3\) in the definition of conjugates. What do you get? (You should get that \(a = 0\) or \(b = 0\), but \(b = 0 \) is impossible, so \(a = 0\)).
   3. Finally, conclude that there is no rational number \(b\) such that \(\sqrt 3 = b \sqrt 2\).

2. Here is a version of the above problem that might help in the homework. Let \(\zeta_3 = e^{2\pi i/3}\). For \(a,b,c \in \mathbf{Q}\), prove that \(a + b 2^{1/3} + c 2^{2/3}\) and \(a + b \zeta_3 2^{1/3} + \zeta_3^2 2^{2/3}\) are conjugates over \(\mathbf{Q}\).

   Hint: Use the presentation \(\mathbf{Q}[t]/(t^3-2)\) for \(\mathbf{Q}[2^{1/3}]\) and \(\mathbf{Q}[2^{1/3} \zeta_3]\).

3. Let \(\gamma = \sqrt 2 + \sqrt 3\).
   1. Prove that \(\mathbf{Q}[\gamma] = \mathbf{Q}[\sqrt 2, \sqrt 3]\). Call this field \(K\).
   2. Prove that \(1, \gamma, \gamma^2, \gamma^3\) and \(1, \sqrt 2, \sqrt 3, \sqrt 6\) are both bases of \(K\) as a \(\mathbf{Q}\)-vector space. Write down the change-of-basis matrix.

Homework 4 :: Due on Friday 21 Mar 2025 by 11:59pm on Gradescope.

1. Exercise 15.3.5
2. Exercise 15.3.7

Homework 5 :: Due on Friday 28 Mar 2025 by 11:59pm on Gradescope.

1. For which \(m, n \in \mathbf{Z}\) is the ring \(\mathbf{Q}[x,y]/(x^2-m, y^2-n)\) a field?
2. Exercise 15.7.8.

Homework 6 :: Due on Friday 18 April 2025 by 11:59pm on Gradescope.

1. Exercise 15.10.1. Recall that an algebraic number in \(\mathbf{C}\) is one that is algebraic over \(\mathbf{Q}\).

2. Exercise 15.M.3. Let us make this question a bit more precise. What are the possible degrees of the irreducible factors of \(f\) in \(K[x]\)? That is, for each \(d \in \{1,\cdots, 6\}\), determine whether \(d\) is possible as a degree of an irreducible factor of \(f \in K[x]\). If you say that a degree \(d\) is possible, please give an example of an \(F\), a \(K\), and an \(f\) such that \(f \in K[x]\) has an irreducible factor of degree \(d\). If you say that a degree \(d\) is not possible, please explain why.

   We can ask for a refinement of this question (but you do not have to do this). The factorisation of \(f \in K[x]\) gives a partition of 6. For example, if \(f\) factors into two linear factors and two quadratic factors, we have the partition \(6 = 1 + 1 + 2 + 2\). What are the possible partitions of 6?

Due on Friday 2 May 2025 by 11:59pm on Gradescope.

The goal of this mini-project is to explore the factorisation of prime numbers in cyclotomic rings.

Let \(\zeta_n = e^{2\pi i / n}\). The \(n\)-th cyclotomic ring is the ring \(\mathbf{Z}[\zeta_n]\).

1. Let \(p\) be an integer prime number. Consider the map \(\mathbf{Z}[x] \to \mathbf{Z}[\zeta_p]\) that sends \(x\) to \(\zeta_p\). Find the kernel of this map. Justify your answer.
2. Let \(\ell\) be an integer prime number. Prove that \(\ell\) is prime in \(\mathbf{Z}[\zeta_p]\) if and only if the polynomial \(x^{p-1} + \cdots + 1\) is irreducible in \(\mathbf{F}_{\ell}[x]\). For \(p = 5\), give examples of integer primes \(\ell\) that are also primes in \(\mathbf{Z}[\zeta_p]\) and those that are not primes in \(\mathbf{Z}[\zeta_p]\). (At least 3 examples of each kind).
3. Suppose \(\ell \neq p\). Prove that \(x^{p-1} + \cdots + 1\) is reducible in \(\mathbf{F}_{\ell}[x]\) if and only if \(p\) divides \(\ell^{n} - 1\) for some positive integer \(n < p-1\). Observe (but do not write-up) that this statement is equivalent to: \(x^{p-1}+\cdots+1\) is irreducible in \(\mathbf{F}_{\ell}[x]\) if and only if the order of \(\ell\) in \(\mathbf{F}_{p}^{\times}\) is \((p-1)\).
4. This problem involves doing experiments using a computer. You may use a program of your choice (my choice is sage, which has many mathematical algorithms built-in). Ask me for help if you need it. You do not need to prove any of your results. As justification for your answers, include a copy of your code or pseudo-code.
   1. Fix a small prime \(p\) (one or two digits). Consider all primes \(\ell\) up to a large number \(N\).
      • What proportion of \(\ell\) are such that \(\ell\) has order \((p-1)\) in \(\mathbf{F}_p^{\times}\)?
      • Where do these numbers converge as \(N \to \infty\)?
      • Does the limiting value seem to depend on \(p\)?
      • (Optional, do not turn in) Can you guess the limiting value in terms of \(p\)? (Hint: it involves Euler’s totient function applied to \(p-1\)).
   2. Fix a small prime \(\ell\) (one or two digits). Consider al primes \(p\) up to a large number \(N\).
      • What proportion of \(p\) are such that \(\ell\) has order \((p-1)\) in \(\mathbf{F}_p^{\times}\)?
      • Where do these numbers seem to converge as \(N \to \infty\)?
      • Does the limiting value seem to depend on \(\ell\)?
      • (Optional, do not turn in) The phenomenon here has to do with an open problem called Artin’s conjecture (named after Emil Artin, the father of Michael Artin, who is the author of our textbook). Look it up on the internet and see if your answer matches the conjecture.

Due on Friday 23 May 2025 by 11:59pm on Gradescope.

The goal of this project is to explore the inverse Galois problem. This problem asks whether given a finite group \(G\), we can find a finite Galois extension \(\mathbf{Q} \subset K\) such that \(\operatorname{Gal}(K/ \mathbf{Q})\) is isomorphic to \(G\).

1. Let \(p\) be a prime number and let \(G = \mathbf{Z}/ (p-1) \mathbf{Z}\). Find an isomorphism \[ G \to \operatorname{Gal}(\mathbf{Q}(\zeta_p)/ \mathbf{Q}).\]

2. Let \(n\) be an integer that divides \(p-1\) for some prime number \(p\). Find a finite Galois extension \(K\) of \(\mathbf{Q}\) such that \(\operatorname{Gal}(K/ \mathbf{Q})\) is isomorphic to \(\mathbf{Z} / n \mathbf{Z}\).

   Given \(n\), there always exists a prime number \(p\) such that \(n\) divides \(p-1\). So the construction above takes care of all cyclic Galois groups.

3. Let us explore a more quantitative version of the inverse Galois problem: for polynomials of degree \(n\), how frequently is the Galois group equal to the full symmetric group \(S_n\)? Gather some numerical data.

   1. Fix \(n\).
   2. Let \(P_{n,N}\) be the set of monic polynomials of degree \(n\) with coefficients in \(\{-N,-N+1, \cdots, N-1, N\}\). What fraction of the elements of \(P_{n,N}\) have Galois group equal to \(S_n\)?
   3. Does there seem to be a limiting value as \(N \to \infty\)?
   4. In (2), checking all polynomials may be computationally infeasible. An alternate approach is to pick a large number of samples from \(P_{n,N}\) uniformly randomly, and to calculate the fraction that have Galois group \(S_n\). If the sample size is large enough, then this fraction should be very close to the value we want.

   Based on the data, can you make a conjecture?

   In your response, include

   1. the code/pseudo-code that you used to generate the data,
   2. the data itself,
   3. any conjecture(s).

   You may use the following piece of sage code that checks whether a given degree \(n\) polynomial has Galois group \(S_{n}\).

   S = QQ[x]
   def is_galois_group_full(p):
       n = S(p).degree()
       return S(p).is_irreducible() and S(p).galois_group().order() == factorial(n)
   
   # Tests
   is_galois_group_full(x^3-3*x+1) #false
   is_galois_group_full(x^4-1) #false
   is_galois_group_full(x^3+3*x+1) #true
   

Homework 7 :: Due on Friday 25 Apr 2025 by 11:59pm on Gradescope.

1. Prove that the map \[ \mathbf{Q}[x,y]/(x^2-2, y^3-3) \to \mathbf{Q}[\sqrt 2, \sqrt[3]{3}]\] that sends \(x\) to \(\sqrt 2\) and \(y\) to \(\sqrt[3]{3}\) (and is the identity on \(\mathbf{Q}\)) is an isomorphism.

2. Describe all automorphisms of the field \(\mathbf{Q}[\sqrt 2, \sqrt[3]{3}]\). (An automorphism is an isomorphism to itself).

   Recall that any automorphism must be the identity on \(\mathbf{Q}\) (convince yourself that this is the case, but do not include its proof in your write-up.)

The goal of this workshop is to see the connection between field automorphisms and algebraic relations satisfied by the roots as alluded in the lecture.

Let \(\alpha_1, \dots, \alpha_4\) be the roots of \(x^4-2 = 0\). Let \(K = \mathbf{Q}[\alpha_1, \dots, \alpha_4] \subset \mathbf{C}\).

1. Prove that any automorphism of \(K\) must permute the \(\alpha_i\).
2. Using the above, conclude that we have an injective map \(\operatorname{Aut}(K) \to S_4\).

3. What is the image of this map? That is, which permutations of the roots arise from automorphisms of the field?

   The answer is closely tied to polynomial relations among the roots. Suppose \[\alpha_1 = 2^{1/4}, \quad \alpha_2 = 2^{1/4} i, \quad \alpha_3 = -2^{1/4}, \quad \alpha_4 = -2^{1/4}i.\] Observe that we have the relation \(\alpha_1 + \alpha_3 = 0\) but not the relation \(\alpha_1 + \alpha_{4} = 0\). Conclude that the polynomial \((34)\) cannot be in the image of \(\operatorname{Aut}(K)\).

4. Let us find all possible polynomial relations in the roots. This is simply the kernel of the map \[ \mathbf{Q}[x_1, \dots, x_4] \to K\] that sends \(x_i\) to \(\alpha_i\). Let us ask the computer to find it:

   K = QQ[i+2^(1/4)]
   S.<x1,x2,x3,x4> = PolynomialRing(QQ,4)
   f = S.hom([2^(1/4), i*2^(1/4), -2^(1/4), -i*2^(1/4)],K)
   f.kernel()
   

   Ideal (x2 + x4, x1 + x3, x3^2 + x4^2, x4^4 - 2)
   

   Now that you know the kernel, rule out some other permutations.

5. It is more efficient to find \(\operatorname{Aut} K\) using another presentation. We have the presentation \(\mathbf{Q}[x,y]/(x^2+1, y^4-2) \to K\) given by \(x \mapsto i\) and \(y \mapsto 2^{1/4}\). (Assume for now that this map is an isomorphism, and later convince yourselves that this is indeed the case). Use this presentation to conclude that \(\operatorname{Aut} K\) has 8 elements. Describe them by how they act on \(i\) and \(2^{1/4}\). Use this description to pin down the 8 permutations of \(\alpha_1, \dots, \alpha_{4}\) arising from \(\operatorname{Aut} K\).

Homework 8 :: Due on Friday 2 May 2025 by 11:59pm on Gradescope.

1. Exercise 16.3.2: (b) and (c)

2. Let \(\alpha_1, \alpha_2, \alpha_3, \alpha_4\) be the four complex roots of the irreducible polynomial \(x^4-4x+2 \in \mathbf{Q}[x]\). Find the minimal polynomial of \(\alpha_1\alpha_2+\alpha_3\alpha_4\) over \(\mathbf{Q}\).

   For this problem, you may need to rewrite symmetric functions in terms of elementary symmetric functions. See the workshop for a quick tutorial.

1. Find the degrees of the splitting fields of:
   1. \(x^3-2\) over \(\mathbf{Q}\)
   2. \(x^3-t\) over \(\mathbf{C}(t)\)
2. In all the examples above, let \(K/F\) be the splitting field. Find a nice presentation of \(K\) over \(F\). (That is, write \(K = F[x_1, \dots, x_n] / I\) for an explicit ideal \(I\).)
3. We can express any symmetric expression in the roots in terms of the coefficients, which allows us to find its precise value. For an asymmetric expression, we can find the value up to a finite ambiguity. Here is an example. Let \(\alpha_1, \alpha_2, \alpha_3\) be the roots of \(f(x) = x^3+2x-2\), and let \(\alpha = \alpha_1\alpha_2-\alpha_3\). The \(S_3\) orbit of the expression \(x_1x_2-x_3\) contains 2 other expressions: \(x_1x_3-x_2\) and \(x_2x_3-x_1\). Let \(\beta = \alpha_1\alpha_3 - \alpha_2\) and \(\gamma = \alpha_2\alpha_3-\alpha_1\).
   1. Prove that \((x-\alpha)(x-\beta)(x-\gamma)\) has coefficients in \(\mathbf{Q}\).

   2. Find these coefficients. You have then narrowed down \(\alpha\) to 3 possible values.

      To do this, you will have to rewrite symmetric polynomials in terms of elementary symmetric polynomials. Here is an example of how to do that in sage. Remember that you can copy/paste and evaluate sage code at https://sagecell.sagemath.org/.

      S.<x,y,z> = PolynomialRing(QQ, 3)
      E = SymmetricFunctions(QQ).elementary()
      E.from_polynomial((x*y-z)*(x*z-y)*(y*z-x))  #Modify for your polynomial.
      

      e[2, 2] - e[3] - 2*e[3, 1] - e[3, 1, 1] + 2*e[3, 2] + e[3, 3] +
      2*e[4] + e[4, 1] - 2*e[4, 2] - 5*e[5] + 2*e[5, 1] - 2*e[6]
      

Explanation of notation:

• \(e[i]\) is the \(i\)-th elementary symmetric polynomial \(e_i\).
• \(e[i,j]\) is \(e_i \cdot e_j\), and so on.
• Since we only have 3 variables, \(e_{4}\) and above are zero. So, the expression sage gave reduces to \[e_2^2 - e_3 - 2 e_3e_1 + e_3e_1^2 + 2 e_3e_2 + e_3^2.\]

You don’t have to use sage. Feel free to use whatever method (including computation by hand). And, of course, I do not expect you to do this kind of heavy computation on an exam.

The goal of the workshop is to see the main theorem of Galois theory in action in a moderately complicated example.

Let \(K = \mathbf{Q}[3^{1/4},i]\). This is the splitting field of \(x^4-3\), and has the presentation \[ \mathbf{Q}[x,y]/(x^4-3, y^2+1) \to K,\] where the map above sends \(x\) to \(3^{1/4}\) and \(y\) to \(i\).

1. Use the presentation above to find \(\operatorname{Aut}(K/ \mathbf{Q})\).
2. Find an isomorphism from the dihedral group \(D_4\) to \(\operatorname{Aut}(K / \mathbf{Q})\).
3. Write the subgroup diagram for \(D_4\): a vertex for each subgroup and an arrow for each inclusion.
4. For each group in the subgroup diagram, find the fixed field.

The main theorem of Galois theory says that these are all the fields containing \(\mathbf{Q}\) and contained in \(K\).

Homework 9 :: Due on Friday 7 May 2025 by 11:59pm on Gradescope.

1. Let \(p\) be a prime number and let \(F = \mathbf{Q}[e^{2\pi i /p}]\). Let \(K\) be the splitting field of \(x^p-2\) over \(F\). Describe all elements of \(\operatorname{Aut}(K/F)\). Describe a group isomorphism \(\mathbf{Z}/p \mathbf{Z} \to \operatorname{Aut}(K/F)\).

2. Consider the automorphisms \(\sigma\)and \(\tau\) of \(\mathbf{C}(t)\) that fix \(\mathbf{C}\) and act on \(t\) as follows: \[ \sigma (t) = e^{2\pi i/n} t, \text{ and } \tau (t) = t^{-1}.\] Then \(\sigma\) and \(\tau\) generate the dihedral group \(D_{n}\). (Convince yourselves that this is the case, but you do not have to include a proof in your write-up.)

   Find the fixed field \(\mathbf{C}(t)^{D_n}\).

Homework 10 :: Due on Friday 16 May 2025 by 11:59pm on Gradescope.

1. Find an example, with justification, of fields \[ F \subset K \subset L,\] such that \(F \subset K\) and \(K \subset L\) are both Galois extensions, but \(F \subset L\) is not a Galois extension.

2. Let \(\alpha \in \mathbf{C}\) be a root of \(p(x) = x^4 + x + 1\). It turns out that \(p(x) \in \mathbf{Q}[x]\) is irreducible and the Galois group of its splitting field is \(S_4\) (you do not have to prove either of these).

   Prove that there is no field \(K\) with \[\mathbf{Q} \subsetneq K \subsetneq \mathbf{Q}(\alpha).\]

The goal of this workshop is to use Galois theory to settle the question of ruler/compass constructions. We say that \(x \in \mathbf{C}\) is constructible if it lies in a tower of quadratic extensions. That is, if there exist fields \[ \mathbf{Q} \subset K_1 \subset K_2 \subset \cdots \subset K_n,\] where each \(K_i \subset K_{i+1}\) is a degree 2 extension and \(\alpha \in K_n\). Let us prove the following.

\bigskip

\noindent Theorem. A number \(x \in \mathbf{C}\) is constructible if and only if its Galois group (= the Galois group of the splitting field of its minimal polynomial) has order \(2^m\) for some \(m\).

\bigskip

Let us prove the if part. Suppose \(z\) is constructible. It is enough to construct a Galois extension (= splitting field) that contains \(z\) and whose degree over \(\mathbf{Q}\) is a power of 2.

1. Suppose \(z = \sqrt {2 + \sqrt 3}\). Construct a Galois extension (= splitting field) that contains \(z\) and whose degree over \(\mathbf{Q}\) is a power of 2.
2. Do the same for \(\sqrt{2 + \sqrt {3 + \sqrt 5}}\).
3. Try to formalise the pattern you see. This is a bit tricky. One clean way is to prove the following by induction: there exists field \(K'_i\) containing \(K_i\) such that (a) \(K'_i/ \mathbf{Q}\) is Galois and (b) the degree of \(K'_i / \mathbf{Q}\) is a power of 2.

Let us now try the converse. Suppose \(z\) lies in a Galois extension of degree \(2^{m}\) over \(\mathbf{Q}\). Let \(G\) be the Galois group.

1. Here is a fact from group theory: a group of order \(2^m\) has a subgroup of order \(2^{m-1}\). Use it to produce a chain of subgroups of size \(1,2,\cdots, 2^m\).
2. Apply the main theorem of Galois theory.

Homework 11 :: Due on Friday 23 May 2025 by 11:59pm on Gradescope.

1. By following the procedure in the book for the quintic, find a polynomial of degree \(7\) in \(\mathbf{Q}[x]\) whose Galois group is \(S_7\).

2. A simple group is a group that has no normal subgroups other than the identity and the group itself. For example, for a prime \(p\), the group \(\mathbf{Z}/p \mathbf{Z}\) is simple, and these are the only simple abelian groups. There are many non-abelian finite simple groups. In a massive undertaking in the later half of the last century, many mathematicians working together found out all finite simple groups. The picture below (from wikipedia) lists them.

   Let \(f(x) \in \mathbf{Q}[x]\) be a polynomial whose Galois group over \(\mathbf{Q}\) is a non-abelian simple group \(G\). Let \(\mathbf{Q} \subset K\) be a Galois extension with Galois group \(\mathbf{Z}/ p \mathbf{Z}\). Prove that the Galois group of \(f(x)\) over \(K\) is also \(G\).