v1-build.33.md

1st Edition, Build 33

General Content Changes

• Fixed a minor error in the title of the book.
• Changed page size to be roughly executive page size (7" x 10") with a 0.25" bleed margin.

Part 0 (Preliminaries) Changes

• Fixed minor inconsistency in capitalisation in section 2.2.
• Made a small change to the statement of exercise 2.5.
• Fixed inconsistency in capitalisation of statement variables in definition 2.4.4 and definition 2.4.6.
• Fixed typo in section 3.1.
• Removed remark about the initialism "BWOC" in section 3.4.
• Improved the proofs in example 3.4.3 and example 3.5.3.
• Made the proof in example 3.5.4 clearer.
• Fixed confusing remark after axiom 4.1.7.
• Rewritten the examples in subsection 4.3.1.
• Made definition 5.1.1 clearer.
• Fixed examples in section 5.3 to correctly use the function definition.
• Clarified definition 5.3.7 and definition 5.3.11.
• Fixed incorrect definition of function composition in definition 5.5.1.
• Made axiom 5.5.2 clearer.
• Updated definition 5.6.1.
• Fixed example 5.6.5.
• Reordered the final part of section 5.6.
• Made other minor changes to the text.

Part I (Groups) Changes

• Made example 8.1.5 and example 8.1.7 more concise.
• Edited propositions' statements and proofs in section 8.2.
• Slightly edited example 8.3.3.
• Edited the proof of proposition 8.4.4 and theorem 8.4.5.
• Added labels to the vertices in figure 8.1.
• Added more detail to example 8.5.4 and example 8.5.5.
• Made the statement of problem 8.7 clearer.
• Edited definition 9.1.1 and its remark to be nicer.
• Fixed minor issues in example 9.1.2.
• Fixed minor typo in definition 9.1.3.
• Made the proof of theorem 9.2.1 (subgroup test) more concise.
• Made example 9.2.2 clearer.
• Made definition 9.3.1 clearer.
• Updated the statement and proof of lemma 9.3.3 (coset equality).
• Slightly changed the remark after corollary 9.3.4.
• Made the proof of lemma 9.4.4 more succinct.
• Rewritten the proof of theorem 9.3.5 (Lagrange).
• Made the proof of corollary 9.4.6 less confusing.
• Reworded the statement of corollary 9.4.8.
• Reworded the statement of exercise 9.7.
• Rewritten the starting part of section 9.6.
• Edited the definition of the groups mentioned in example 10.1.3.
• Made some minor edits to proposition 10.2.1 and proposition 10.2.2.
• Clarified definition 10.3.1 and definition 10.3.2.
• Edited the proof of theorem 10.5.2 and theorem 10.5.3.
• Made example 11.1.1 clearer.
• Rewritten the statement for exercise 11.2.
• Edited the statement for corollary 11.2.6.
• Edited the proof for theorem 11.3.1 (Cayley's theorem).
• Clarified the statement of proposition 12.1.2.
• Made some minor edits to the proof of theorem 12.1.6.
• Added a new remark after problem 12.4.
• Changed "Fundamental Homomorphism Theorem" to "First Isomorphism Theorem".
• Changed "Diamond Isomorphism Theorem" to "Second Isomorphism Theorem".
• Added more detail for example 13.5.2.
• Fixed a weird space for the symbol of the General/Special linear group of real numbers.
• Made minor changes to the names of section 14.2, section 14.3, subsection 14.5.2, subsection 14.5.3, subsection 14.6.1, and subsection 14.6.2.
• Edited definition 14.4.1 slightly.
• Made a minor edit to axiom 14.4.5.
• Edited the proof of proposition 14.5.2 and proposition 14.5.4.
• Rewritten the proof of proposition 14.6.7.
• Edited part of the statement of problem 14.4.
• Fixed a small issue with definition 15.1.1.
• Clarified the symbol $g$ used in lemma 15.4.2 (Burnside).
• Made the remark after definition 15.5.2 more succinct.
• Cleaned up the proof of theorem 15.6.1 (Class Equation).
• Made the statement of problem 15.3 clearer.
• Edited the introduction to chapter 16 to flow better.
• Moved exercise 16.1 to a more suitable location.
• Made definition 16.3.1 more airtight.
• Made the statement of exercise 16.5 clearer.
• Changed the remark after proposition 16.3.3 to be clearer.
• Updated the proof of proposition 16.3.4 to be clearer.
• Removed remark after the proof of theorem 16.4.1 (Sylow II).
• Fixed titling format for section 16.6.
• Added more detail about the Third Sylow Theorem in section 16.5.
• Made definition 16.6.1 clearer.
• Added additional conditions in the statement of problem 16.5.
• Fixed issues in the proofs of lemma 17.3.1, lemma 17.3.2, and lemma 17.3.4.
• Fixed missing reference in the proof of lemma 17.3.2.
• Fixed introduction to chapter 18.
• Updated definition 18.1.1, definition 18.1.3, and definition 18.2.1 to be clearer.
• Fixed a minor issue with the second remark after definition 18.3.1.
• Added a new example in section 18.3 to explain why the group of integers has no composition series.
• Edited example 18.5.2 to be clearer.
• Added more detail in the proof of theorem 18.5.3 (Schreier Refinement Theorem).
• Made the proof of theorem 19.2.1 more succinct.
• Edited the proof of theorem 19.2.4 to be clearer.
• Rewritten the statement of corollary 19.2.5.
• Made other minor changes to the text.

Part II (Rings) Changes

• Changed the preamble to chapter 20.
• Rewritten definition 20.1.1, definition 20.1.2, definition 20.1.3, and definition 20.2.1 to be more concise.
• Fixed a minor typo in the proof of proposition 21.1.10.
• Made the proof of proposition 21.3.6 clearer.
• Edited definition 21.4.1 to be clearer.
• Rewritten the statements of problem 21.4 and problem 21.8 to be clearer.
• Added text in between definition 22.3.1 and definition 22.3.2 to make it clear about the intentions of the definitions.
• Fixed typos in the proof of proposition 22.3.4.
• Fixed a small mistake in the introduction to chapter 23.
• Fixed a critical error at the beginning of section 23.1.
• Edited the explanation after the remark of definition 23.2.1.
• Modified definition 23.2.2 to be clearer.
• Changed the symbols used for ideals in section 23.3.
• Updated the statement of exercise 23.4 to be more succinct.
• Changed definition 23.5.1 and the description after definition 23.5.1 slightly.
• Fixed a slight grammatical error in example 23.5.4.
• Made exercise 23.10 more succinct.
• Added a new example (example 23.6.3) about the annihilator.
• Fixed error in the proof of proposition 23.6.8.
• Made definition 24.1.1 more airtight.
• Added more detail to the statement of exercise 24.1.
• Rewritten text around definition 24.1.6.
• Fixed grammar in definition 25.1.1.
• Rewritten the proof of proposition 25.1.5 to be more succinct.
• Fixed a minor typo in example 25.1.10.
• Edited theorem 25.3.3 to be clearer.
• Rewritten theorem 25.4.1 (polynomial long division), corollary 25.4.6 (remainder theorem), and corollary 25.4.7 (factor theorem) to be more succinct.
• Made the statement of exercise 26.2 clearer.
• Fixed slight awkwardness of text in definition 26.2.7.
• Cleaned up the proof of lemma 26.2.8.
• Slightly edited the statement and proof of theorem 26.2.9 to be more succinct.
• Made the explanation offered in example 26.2.15 more concise.
• Cleaned up the proof of corollary 26.3.2.
• Fixed erroneous irreducible polynomial in example 26.3.6.
• Fixed typos in example 26.4.4 and example 26.4.5.
• Rewritten definition 27.2.3.
• Edited the proof of proposition 27.2.4 to be clearer.
• Fixed typo in the proof of proposition 27.2.6.
• Made the proof of proposition 27.3.3 more spaced out and clearer.
• Fixed slight awkwardness in the sketch of the proof of theorem 27.4.4.
• Slightly edited the statement of problem 27.14.
• Edited the description of NTRU parameters in subsection 28.3.1.
• Made other minor changes to the text.

Part III (Fields) Changes

• Moved terminology of (in)finite fields into a new definition.
• Made definition 29.3.1 more airtight.
• Fixed slight confusion within definition 30.3.1.
• Edited part of the proof of theorem 31.2.6 (specifically the surjective proof of the second isomorphism) to be less confusing.
• Made example 31.2.14 more concise.
• Changed the statement of theorem 31.2.16 to be clearer.
• Fixed slight error in the proof of theorem 31.2.16.
• Amended part of the proof of theorem 31.2.21.
• Added more explanation to the proof of theorem 31.5.3.
• Edited definition 31.5.5 to make the definition of perfect fields clearer.
• Moved exercise 31.9 to the end of section 31.5.
• Edited the preamble to chapter 32 to be less awkward and more grammatically sound.
• Modified text before definition 32.2.3 to be clearer.
• Slightly edited example 32.3.3 to improve clarity.
• Edited the proof of corollary 32.4.7 to be more concise.
• Changed incorrect term used in the text before theorem 33.1.3.
• Edited part of example 33.1.6 to have a more consistent style with the rest of the text.
• Modified definition 34.1.6 to be less verbose.
• Added emphasis for the terms "similar" and "congruent" found in definition 34.1.15 and definition 34.1.17.
• Made other minor changes to the text.

Part IV (Galois) Changes

• Updated the preamble to the part.
• Edited the proofs of proposition 35.1.1 and proposition 35.1.2 to be clearer.
• Added a caption to the tables in example 35.1.5.
• Modified the proof of corollary 35.1.12.
• Edited the statement of corollary 35.5.3 to be clearer.
• Fixed minor errors in example 35.5.5.
• Fixed the fact before proposition 35.7.2.
• Added a label for the graph of the quintic in proposition 35.7.2.
• Modified the statements of definition 36.1.1, proposition 36.1.2, definition 36.1.4, definition 36.1.5, and proposition 36.1.9 to be clearer and more precise.
• Fixed the article used in the text before definition 36.1.4.
• Edited awkward last sentence before axiom 36.1.8.
• Removed awkward phrase "the calculus inclined" in the text before 36.2.2.
• Slightly edited the statement of lemma 36.3.2.
• Made other minor changes to the text.

Other Changes

• Cleaned up the code for tables.