No changes.
- Added a new axiom 5.5.2 on the associativity of function composition.
- Added a new problem 5.8 on how an injective function from a finite set to itself is a bijection.
- Clarified definition of the centralizer in example 9.2.3.
- Amended citation to coset equality lemma before corollary 9.3.4.
- Fixed slight issue with wording in proof of lemma 9.4.3.
- Changed numbered points to bullet points in the proof of corollary 9.4.6.
- Cleaned up the organization of definition 9.6.1.
- Tidied up proof of theorem 9.6.2.
- Tidied up wording in example 9.6.3.
- Edited wording of problem 9.3 and problem 9.9 to make it clearer and more precise.
- Added remarks on the etymology of homomorphisms and isomorphisms.
- Updated definition 10.1.1 to be more consistent with the more recently added definitions.
- Changed homomorphism symbol used in example 10.1.3.
- Made the preamble to section 10.2 less awkward.
- Changed the wording of proposition 10.2.3 to be more precise.
- Fixed formatting issues and typos in the proof of proposition 10.2.3.
- Added missing full stop before proposition 10.2.4.
- Edited style of the proof of proposition 10.2.6.
- Edited style of example 10.3.3 and example 10.3.4 to be more consistent with more recent examples.
- Removed italics from a phrase in the preamble of section 10.4.
- Rephrased exercise 10.10 to use the language of equivalence relations.
- Edited style of the proof to theorem 10.5.2 and 10.5.3.
- Changed wording of problem 10.2.
- Cleaned up the preamble for chapter 11.
- Edited introduction for section 11.1 to be more consistent with formatting of text in later chapters.
- Edited remark after example 11.1.1.
- Made the use of "cycle notation" more consistent in example 11.1.6, example 11.1.7, and exercise 11.2.
- Reworded proposition 11.2.2.
- Modified the proof of proposition 11.2.2 to use axiom 5.5.2.
- Changed definition 11.2.3 to include the alternate name for the symmetric group of degree
$n$ . - Added a new proposition and exercise explicitly proving that the external direct product is indeed a group.
- Changed the notation used in example 12.1.4 to be more consistent with the notation of the multiplicative group of fields.
- Cleaned up the proof of proposition 12.1.5 to be less confusing.
- Modified the remark after proposition 12.2.2.
- Amend the proof of the "well-definedness" of
$\phi$ in the proof of theorem 12.3.1. - Fixed description of exercise 12.3.2.
- Made "Klein four group" bolded in problem 12.4.
- Edited remark after definition 13.1.1 and definition 13.2.1 to increase clarity.
- Edited proof of proposition 13.1.4 to be more concise.
- Reworded example 13.2.2 to be clearer.
- Edited exercise 13.2 to be clearer.
- Clarified remark after theorem 13.3.1.
- Edited text after the statement of theorem 13.3.1.
- Clarified wording of "multiplication" in example 13.3.2.
- Changed description of the subgroup lattice for theorem 13.4.1.
- Removed single word in example 13.4.3.
- Removed some text in example 13.4.4.
- Clarified what "it" refers to in the proof of statement 2 of theorem 13.5.1.
- Rewrote lemma 14.3.2, its proof, and its remark to be more concise and clearer.
- Rewrote lemma 14.3.4 to be clearer.
- Modify the definition of an inversion in definition 14.3.5.
- Removed notation for inversion after definition 14.3.5.
- Modify example 14.3.6 to accommodate the new changes.
- Reworded the motivation for theorem 14.3.8.
- Made description after exercise 14.7 more concise.
- Slightly modify the proof of proposition 14.3.12.
- Rewritten proposition 14.3.13 and its proof to be clearer.
- Move the expression of Euler's totient function into definition 14.4.3.
- Added condition that
$a < n$ into definition 14.4.4. - Modified the proof of proposition 14.4.8 to be more concise.
- Added small proof that the identity matrix is its inverse to subsection 14.5.1.
- More concisely define the matrix determinant in subsection 14.5.1.
- Added statement of proposition that claims that the general linear group is indeed a group.
- Slightly change the text before proposition 14.5.4.
- Added missing brackets between the determinant function in the proof of proposition 14.5.4.
- Made the notation for the multiplicative group of real numbers in subsection 14.5.4 consistent with future chapters.
- Renamed subsections 14.6.1 and 14.6.2, removing the redundant "of
$G$ ". - Added statement of proposition that claims that the group of automorphisms is indeed a group.
- Fixed some grammar in the proof that the group of automorphisms is indeed a group.
- Fixed some grammar issues in definition 14.6.5.
- Made the statements of proposition 14.6.6 and proposition 14.6.7 more precise.
- Fixed a small typo in the sketch of the proof of theorem 16.1.2.
- Added new proposition 22.2.13 on how fields have no proper ideals.
- Changed the problem that proves theorem 28.2.11.
- Added a new chapter 30 on extension fields and splitting fields.
- Added correct PDF metadata.
- Moved the "work in progress" watermark from the part on rings to the part on fields.