Solutions to exercises from *Abstract Algebra* by Dummit & Foote. (Not all links are active.)

- Chapter 0: Preliminaries
- Chapter 1: Introduction to Groups
- Chapter 2: Subgroups
- Chapter 3: Quotient Groups and Homomorphisms
- Chapter 4: Group Actions
- Chapter 5: Direct and Semidirect Products and Abelian Groups
- Chapter 6: Further Topics in Group Theory
- Chapter 7: Introduction to Rings
- Chapter 8: Euclidean Domains, Principal Ideal Domains, and Unique Factorization Domains
- Chapter 9: Polynomial Rings
- Chapter 10: Introduction to Module Theory
- Chapter 11: Vector Spaces
- Chapter 12: Modules over Principal Ideal Domains
- Chapter 13: Field Theory
- Chapter 14: Galois Theory
- Chapter 15: Commutative Rings and Algebraic Geometry
- Chapter 16: Artinian Rings, Discrete Valuation Rings, and Dedekind Domains
- Chapter 17: Introduction to Homological Algebra and Group Cohomology
- Chapter 18: Representation Theory and Character Theory
- Chapter 19: Examples and Applications of Character Theory
- Appendix I: Cartesian Products and Zorn’s Lemma
- Appendix II: Category Theory

## Solutions

**Chapter 0: Preliminaries****Chapter 1: Introduction to Groups**- §1.1: Basic Axioms and Examples
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36)
- §1.2: Dihedral Groups
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18)
- §1.3: Symmetric Groups
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20)
- §1.4: Matrix Groups
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11)
- §1.5: The Quaternion Group
- (#1) (#2) (#3)
- §1.6: Homomorphisms and Isomorphisms
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26)
- §1.7: Group Actions
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23)

**Chapter 2: Subgroups**- §2.1: Definition and Examples
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17)
- §2.2: Centralizers and Normalizers, Stabilizers and Kernels
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14)
- §2.3: Cyclic Groups and Cyclic Subgroups
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26)
- §2.4: Subgroups Generated by Subsets of a Group
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20)
- §2.5: The Lattice of Subgroups of a Group
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20)

**Chapter 3: Quotient Groups and Homomorphisms**- §3.1: Definitions and Examples
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- §3.2: More on Cosets and Lagrange’s Theorem
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23)
- §3.3: The Isomorphism Theorems
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10)
- §3.4: Composition Series and the Hölder Program
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12)
- §3.5: Transpositions and the Alternating Group
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17)

**Chapter 4: Group Actions**- §4.1: Group Actions and Permutation Representations
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10)
- §4.2: Groups Acting on Themselves by Left Multiplication – Cayley’s Theorem
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14)
- §4.3: Groups Acting on Themselves by Conjugation – The Class Equation
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36)
- §4.4: Automorphisms
- §4.5: The Sylow Theorems
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) (#41) (#42) (#43) (#44) (#45) (#46) (#47) (#48) (#49) (#50) (#51) (#52) (#53) (#54) (#55) (#56)
- §4.6: The Simplicity of
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8)

**Chapter 5: Direct and Semidirect Products and Abelian Groups**- §5.1: Direct Products
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18)
- §5.2: The Fundamental Theorem of Finitely Generated Abelian Groups
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16)
- §5.3: Table of Groups of Small Order
- (#1)
- §5.4: Recognizing Direct Products
- §5.5: Semidirect Products
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25)

**Chapter 6: Further Topics in Group Theory**- §6.1: p-Groups, Nilpotent Groups, and Solvable Groups
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38)
- §6.2: Applications in Groups of Medium Order
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30)
- §6.3: A Word on Free Groups
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14)

**Chapter 7: Introduction to Rings**- §7.1: Basic Definitions and Examples
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30)
- §7.2: Examples: Polynomial Rings, Matrix Rings, and Group Rings
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13)
- §7.3: Ring Homomorphisms and Quotient Rings
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37)
- §7.4: Properties of Ideals
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) (#41)
- §7.5: Rings of Fractions
- (#1) (#2) (#3) (#4) (#5) (#6)
- §7.6: The Chinese Remainder Theorem
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11)

**Chapter 8: Euclidean Domains, Principal Ideal Domains, and Unique Factorization Domains****Chapter 9: Polynomial Rings**- §9.1: Definitions and Basic Properties
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18)
- §9.2: Polynomial Rings over Fields I
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13)
- §9.3: Polynomial Rings that are Unique Factorization Domains
- (#1) (#2) (#3) (#4) (#5)
- §9.4: Irreducibility Criteria
- §9.5: Polynomial Rings over Fields II
- (#1) (#2) (#3) (#4) (#5) (#6) (#7)
- §9.6: Polynomials in Several Variables over a Field and Gröbner Bases
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) (#41) (#42) (#43) (#44) (#45)

**Chapter 10: Introduction to Module Theory**- §10.1: Basic Definitions and Examples
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23)
- §10.2: Quotient Modules and Module Homomorphisms
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14)
- §10.3: Generation of Modules, Direct Sums, and Free Modules
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27)
- §10.4: Tensor Products of Modules
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- §10.5: Exact Sequences – Projective, Injective, and Flat Modules
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**Chapter 11: Vector Spaces**- §11.1: Definitions and Basic Theory
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14)
- §11.2: The Matrix of a Linear Transformation
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39)
- §11.3: Dual Vector Spaces
- (#1) (#2) (#3) (#4) (#5)
- §11.4: Determinants
- (#1) (#2) (#3) (#4) (#5) (#6)
- §11.5: Tensor Algebras, Symmetric and Exterior Algebras
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14)

**Chapter 12: Modules over Principal Ideal Domains**- §12.1: The Basic Theory
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22)
- §12.2: The Rational Canonical Form
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25)
- §12.3: The Jordan Canonical Form
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) (#41) (#42) (#43) (#44) (#45) (#46) (#47) (#48) (#49) (#50) (#51) (#52) (#53) (#54) (#55) (#56) (#57) (#58) (#59)

**Chapter 13: Field Theory**- §13.1: Basic Theory of Field Extensions
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8)
- §13.2: Algebraic Extensions
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22)
- §13.3: Classical Straightedge and Compass Constructions
- (#1) (#2) (#3) (#4) (#5)
- §13.4: Splitting Fields and Algebraic Closures
- (#1) (#2) (#3) (#4) (#5) (#6)
- §13.5: Separable and Inseparabel Extensions
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11)
- §13.6: Cyclotomic Polynomials and Extensions
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17)

**Chapter 14: Galois Theory**- §14.1: Basic Definitions
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10)
- §14.2: The Fundamental Theorem of Galois Theory
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31)
- §14.3: Finite Fields
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17)
- §14.4: Composite Extensions and Simple Extensions
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8)
- §14.5: Cyclotomic Extensions and Abelian Extensions over
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18)
- §14.6: Galois Groups of Polynomials
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) (#41) (#42) (#43) (#44) (#45) (#46) (#47) (#48) (#49) (#50) (#51)
- §14.7: Solvable and Radical Extensions: Insolvability of the Quintic
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21)
- §14.8: Computation of Galois Groups over
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10)
- §14.9: Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19)

**Chapter 15: Commutative Rings and Algebraic Geometry**- §15.1: Noetherian Rings and Affine Algebraic Sets
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) (#41) (#42) (#43) (#44) (#45) (#46) (#47) (#48)
- §15.2: Radicals and Affine Varieties
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) (#41) (#42) (#43) (#44) (#45) (#46) (#47) (#48) (#49) (#50) (#51) (#52) (#53) (#54)
- §15.3: Integral Extensions and Hilbert’s Nullstellensatz
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28)
- §15.4: Localization
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40)
- §15.5: The Prime Spectrum of a Ring
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**Chapter 16: Artinian Rings, Discrete Valuation Rings, and Dedekind Domains**- §16.1: Artinian Rings
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14)
- §16.2: Discrete Valuation Rings
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9)
- §16.3: Dedekind Domains
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25)

**Chapter 17:Introduction to Homological Algebra and Group Cohomology**- §17.1: Introduction to Homological Algebra – Ext and Tor
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35)
- §17.2: The Cohomology of Groups
- §17.3: Crossed Homomorphisms and
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15)
- §17.4: Group Extensions, Factor Sets, and
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10)

**Chapter 18: Representation Theory and Character Theory**- §18.1: Linear Actions and Modules over Group Rings
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24)
- §18.2: Wedderburn’s Theorem and Some Consequences
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18)
- §18.3: Character Theory and the Orthogonality Relations
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28)

**Chapter 19: Examples and Applications of Character Theory**- §19.1: Characters of Groups of Small Order
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17)
- §19.2: Theorems of Burnside and Hall
- (#1) (#2) (#3) (#4) (#5)
- §19.3: Introduction to the Theory of Induced Characters
- (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15)

**Appendix I: Cartesian Products and Zorn’s Lemma****Appendix II: Category Theory**

## Comments

Hi, I really admire your work. Can I get your e-mail address please? I have something to discuss with you, please!

Best Wishes

hello¡¡¡¡

I would like you to help me understand these exercises thanks.

questions

– If a ring has zero divisors I can enlarge to a field F

– If two entire domains D1 and D2 are isomorphic, fields of quotients

are also isomorphic.

– If fi is an isomorphism of fields F1 F2 —– yu units belonging to F1 then the image of u belongs to the units of F2

If fi is an isomorphism of fields F1—— F2 and u belongs units to F1 then the image of u belongs to the units of F2

Simply Put. You are an awesome man/woman!

Thanks a lot🙂 It is really a great work🙂

My Regards and Best Wishes🙂

Excellent work! Really helpful!

Peter

Thanks! I’m glad you find it useful.

Looking forward to some posts on Homology!

Peter

I love you… really, I do. Haha, no, but this “crazy project” is an amazing feat and it’s been extremely helpful. Thanks.

Hi, the links for Sec.12.3 is missing. Could you please fix that so I can learn from it.

Thanks a lot

Hi,

the link for sec12.3 is missing. I can not find any solutions. could you please put them on there.

Thanks

MUY UTIL

Excellent Excellent Excellent work!

Thanks!

I really admire your work.

Best regards,

Thanks for helping me out.🙂

Can i get solutions of Dummit and Foote in pdf or in word form???

unnecessary lemmas. very sloppy. handwriting needs improvement.