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v1-build.29.md

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1st Edition, Build 29

General Content Changes

  • Fixed an issue with copying and pasting text with accents (e.g. Bézout).
  • Updated and cleaned up the interdependence diagram and interdependence notes.

Part 0 (Preliminaries) Changes

  • Added clarification on part (g) of exercise 1.1.
  • Added more description for the solution to exercise 1.2.
  • Made the statement of exercise 2.3 clearer.
  • Changed the brackets used in the solution to exercise 2.6.
  • Fixed incorrect capitalisation of the statement variables in chapter 3.
  • Made the proof in example 3.2.1 clearer.
  • Added missing word in the preamble for section 3.4.
  • Reworded exercise 3.4 to be clearer.
  • Fixed incorrect statement for exercise 4.4.
  • Fixed the solution to exercise 5.6.
  • Added citation for Euclid's division lemma to the solution for exercise 6.6.
  • Clarified notation used in the solution for part (c) of exercise 6.6.

Part I (Groups) Changes

  • Fixed incorrect ordering of the solutions to exercises 8.3 and 8.4 in the solutions section.
  • Slightly changed the statement of exercise 9.8.
  • Added a new chapter on the fundamental theorem of finite abelian groups.

Part II (Rings) Changes

No changes.

Part III (Fields) Changes

  • Added the preamble for part III.
  • Removed poorly defined notation for the order of infinite fields in example 29.1.9.
  • Fixed statement of proposition 29.2.7.
  • Fixed definition of a subfield in definition 29.3.1.
  • Fixed the solution to problem 29.5.
  • Amended definition of a vector space in definition 30.1.1.
  • Fixed explanation for why $(F[x], +)$ is an abelian group in example 30.1.4.
  • Changed the symbols used in theorem 30.1.6 to be more consistent with those used in the rest of the chapter.
  • Modified the description of the subspace test in theorem 30.2.4.
  • Made some changes to the proof of the subspace test.
  • Slightly made definition 30.3.1 more airtight.
  • Rewritten example 30.5.4.
  • Fixed some typos in the proof of theorem 30.5.5.
  • Added more detail on how complex numbers were first defined to example 31.1.1.
  • Removed yet-to-be-introduced notation from example 31.1.4.
  • Made the explanation of the proof of proposition 31.2.3 clearer.
  • Fixed some grammar mistakes in the proof of theorem 31.2.6.
  • Fixed a critical typo in example 31.2.14.
  • Deleted example 31.2.18 because it was wrong.
  • Fixed small typos in the proof of lemma 31.2.21.
  • Made the statement of theorem 31.4.3 clearer.
  • Made the statement of theorem 31.5.3 clearer.
  • Slightly expand the scope of the Freshman's Dream (proposition 31.5.7).
  • Added more detail to section 32.1.
  • Added more results on characterising extensions; specifically, on the minimal polynomial.
  • Split exercise 32.2 into two exercises.
  • Amend the statement of proposition 32.3.2 to be stronger, giving equality instead of just an isomorphism of fields.
  • Added more detail to the proof of theorem 32.5.1.
  • Moved corollary 32.5.7 down one page to make the lone exercise 31.5 less awkward.
  • Fixed a small typo in problem 32.1.
  • Made minor edits to all chapters.
  • Added two new chapters: one on finite fields and one on geometric constructions.

Other Changes

  • Moved symbol definitions from front-matter.tex to a new file symbols.tex.
  • Changed default font encoding from OT1 to T1.
  • Created a new cleanup.py script to clean up source files properly.