\(\newcommand{\from}{\colon}\)\(\newcommand{\cross}{\times}\)\(\newcommand{\cover}[1]{{\widetilde{#1}}}\)\(\newcommand{\CC}{\mathbb{C}}\)\(\newcommand{\QQ}{\mathbb{Q}}\)\(\newcommand{\RR}{\mathbb{R}}\)\(\newcommand{\ZZ}{\mathbb{Z}}\) Questions handed in by students on 2018-10-25. Exercises - 1. Are all fields of the form \( \QQ(\sqrt{p}, \sqrt{q}) \) a simple extension with \( \alpha = \sqrt{p} + \sqrt{q} \)? Answer: Yes. Set \( \beta = \sqrt{p} - \sqrt{q} \) and consider multiplying \( \alpha \) by \( 1 = \beta/\beta \). 2. Is \( \{ \sqrt{p} \mid \mbox{$p$ prime} \) linearly independent over \( \QQ \)? Answer: Yes. This is similar in spirit to Exercise 6.14 of Stewart's book (4th ed.), and he gives hints in Example 6.8. 3. Why do you set so many exercises? I have done all of them but it takes a lot of time! Maybe limit to 1 or 2 per lecture? Answer the first: Please only do the exercises (from lecture) that appeal to you! Answer the second: Some of the in-lecture exercises will make it to the homework. So thought spent early is thought spent wisely... 4. In exercise 1.3 we assumed that \( b^{-1} = 1/b \); why? Answer: I don't understand this question. Exercise 1.3 was about inverting a homomorphism, not inverting an element. 5. How important is it to write assignments up in LaTeX (p.s. I hate LaTeX Answer: I would strongly encourage you to use LaTeX for your assignments. There are many places to get assistance with LaTeX - for example here: https://tex.stackexchange.com/ You are also welcome to ask questions about LaTeX on the forum. 6. Can we assume that any element of \( \QQ(a, b, c) \) can be written as \( \frac{k_1 + k_2 a + k_3 b + k_4 c}{k_5 + k_6 a + k_7 b + k_8 c} \) for some \( k_i \in \QQ \forall i\) and then cancel down? Answer: No, because that is not true. One example is \( \QQ(a, a, a) \) for \( a = \sqrt[100]{2} \). Lecture - Why don't you just write eg for example and ex for exercise? Answer: I didn't think of that. Good suggestion! Terminology - 1. Why do we call scalar multiples of polynomials irreducible? e.g. \( 2x + 2 \in \ZZ[x] \). 2 isn't a unit in \(\ZZ[x]\) so it seems like this is a non-trivial factorization so \( 2x + 2 \) shouldn't e considered irreducible. Answer: In Galois theory the coefficients will (almost) always lie in a field. 2. Consider \( \QQ(\sqrt{2}) \) as an extension of \( \QQ \) I get that we build it using the isomorphism theorem: [much more, omitted]. Answer: I think you are asking about the difference between \( K = \QQ(\sqrt{2}) \), a subfield of \( \RR \), and \( K' = \QQ[x] / (x^2 - 2) \), a quotient of the polynomial ring. These are isomorphic (in two ways) but they are not the same. Now, there are unique monomorphisms \( \iota \from \QQ \to K \) and \( \iota' \from \QQ \to K' \). Both isomorphisms "commute" with these. That is, if \( \sigma \from K' \to K \) is an isomorphism, then \( \sigma \circ \iota' = \iota \). Administration - 1. Will solutions for the assignments be published at some point? Answer: No. There will be worked solutions to other problems given in lecture and in the notes. 2. Is the Maths department so poor that it can't afford giving Galois Theory Notes for free? Answer: If you are in need of financial assistance, please contact Fiona Linton either in person or via email: f dot j dot linton at warwick dot etc 3. What's examinable? (i.e. just lecture material, all lecture notes, all proofs, etc.) Answer: The lecture and homework are the best indicators of the content of the exam. That said, we are closely following Siksek's notes, so those are worth reading and rereading. Folderol - As promised, I have omitted non-academic questions and "questions". :P I also omitted one question that is mathematical, but is definitely not part of Galois theory!