A Web Resource in Mathematics


Studying mathematics on one's own may be a daunting task.
A paradoxical point is that the difficulties one comes across during
self-study may result from an accumulation of minor details inducing
discouragement and sometimes failure.

Thanks to some relevant technical material combined with appropriate
guidelines, it is quite often possible to overcome unduly intimidating
hurdles. 

It does not mean, of course, that all difficulties boil down to these ones.
But this is an example of benefits an efficient though punctual assistance could bring. 

A recurrent problem a student has to solve arises from the multiplicity of textbooks he/she must synthesize.

However well-written they are, textbooks covering the same topics
may have different  prerequisites, different notations, state different
definitions or theorems and to top it off, develop different viewpoints.

Switching back and forth between books in search of solid ground can be a harsh, exasperating experience.

Mathshape's offering consists in the type of assistance just mentioned.
It could be considered as a tailored help meant for the "self-student" in
Mathematics. This is not a consulting service.

You may now want to take a look at the "How it works" page.
How it works....


Requests e-mailed to us should pertain to topics presently covered.
They are listed in relating pages on this site. 

Along with a description of the issues you are facing, we need some
minimal personal information which you could include in your first
message.      

We do our best to examine quickly requests and queries so that we
can let you know, without delay, whether we may be of some help.

If this is the case, we will propose a framework likely to meet with
your approval. Once this framework has been defined, we have to
agree upon a price for the final work but we do not expect any
payment until you are fully satisfied.

This work is forwarded by means of an e-mailed PDF file.

Despite our efforts, we could be unable to meet your needs.
Should it occur, you would owe us nothing.

It is our belief that a sound relationship with our customers, resting on
mutual respect and confidence, as well as conforming with common
sense, can be built without resorting to cumbersome constraints.
Why these topics?



Topics we have selected are not necessarily among the most popular.
However, as a part of the foundations of Algebra, they have crept into
virtually every branch of Mathematics.

For the time being, our focus is Algebra. It conceals so many pitfalls
likely to make a student stumble, that we see a priority in dealing with
its fundamental fields.
Groups
  •  Basic definitions.
  •  Cosets. Index of a subgroup. Lagrange theorem.
  •  Normal subgroups.
  •  Quotient groups and isomorphism theorems.
  •  Cyclic groups.
  •  Permutations (symmetric, alternating, dihedral groups).
  •  Products and sums. Related isomorphisms.
  •  Free group on a set.
  •  Group acting on a set.
  •  Indecomposable groups, Fitting's lemma, Krull-Schmidt theorem.
  •  Sylow subgroups and Sylow theorems.
  •  Solvable groups.
  •  Nilpotent groups.
  •  (Sub)Normal Series. Composition series. Solvable series.
  •   Schreier and Jordan-H�lder theorems.
  •  Free Abelian groups.
  •  Finitely generated Abelian groups.
Rings
  •  Basic definitions.
  •  Ideals.
  •  Isomorphism theorems.
  •  Principal ideal domains.
  •  Prime ideals.
  •  Maximal ideals. Local Rings.
  •  Congruence. Chinese remainder theorem.
  •  Unique factorization domains. Greatest common divisor.
  •  Quotient rings. Quotient fields. Localization.
  •  Polynomials in one or several indeterminates.
  •  Formal power series.
  •  Irreducibility criteria in polynomial rings. 
  •     
Modules
  •  Basic definitions.
  •  Quotient module. Some isomorphism theorems.
  •  Products and sums.
  •  (Short)Exact sequences. The Short Five Lemma.
  •  Split exact sequences.
  •  Free module on a set. Vector spaces. Dimension.
  •  Projective and injective modules.
  •  Groups and modules of homomorphisms. Duality.
  •  Algebras.
  •  Modules over a principal ideal domain.  
Field and Galois Theory
  •  Field extensions: basic definitions and theorems.
  •  Minimal polynomial and related isomorphisms.
  •  Splitting fields and related isomorphisms.
  •  Algebraically closed fields.
  •  Algebraic closure of a field.
  •  Separable extensions.
  •  Normal extensions.
  •  Galois extensions.
  •  Fundamental theorem of Galois theory.
  •  Symmetric rational functions.
  •  Normal closure.
  •  Finite fields.
  •  Purely inseparable extensions.
  •  Separable degree. Inseparable degree.
  •  Primitive element theorem.
  •  Norms and traces.
  •  Abelian and Cyclic extensions.
  •  Hilbert's theorem 90. Decomposition of a cyclic extension.
  •  Prmitive nth roots of unity and related theorems.
  •  Cyclotomic extensions. Cyclotomic polynomials.
  •  Galois group of a polynomial. Basic theorems. Discriminant.
  •  Galois group of a polynomial of degree 3.
  •  Galois group of a polynomial of degree 4. Resolvant cubic.
  •  Radical extensions.
  •  Solvability by radicals.
  •  Roots of the general polynomial of degree 3.
  •  Roots of the general polynomial of degree 4.
  •  The general equation of degree n.
  •  Algebraic independence.
  •  Transcendence bases.
  •  Transcendence degree.
  •  Purely transcendental extensions.
  •  Linear disjointness.
  •  Separating transcendence bases.
  •  Separable extensions (general ca
Multilinear Algebra
  •  
  •  Multilinear mappings and alternating multilinear mappings
  •  Tensor product of modules. Isomorphisms.
  •  N-th exterior power of a module.
  •  Symmetric multilinear mappings.
  •  N-th symmetric power of a module.
  •  Tensor products of homomorphisms.
  •  Direct sums, exact sequences and relating tensor products.
  •  Bimodules and covariant extension.
  •  Flat modules and tensor products.
  •  Quotient ring and tensor products.
  •  
  •  Tensor products of (associative) algebras.
  •  Graded algebras and twisted tensor product.
  •  Anticommutative algebras. Derivations and skew-derivations.
  •  Covariant extension of an algebra.
  •  Generalized derivations and skew-derivations.
  •  Isomorphisms involving twisted tensor products.
  •  Tensor algebra of a module (and of a free module).
  •  Covariant extension of the tensor algebra of a module.
  •  Derivations and skew derivations (on a tensor algebra).
  •  Exterior algebra of a module (and of a free module).
  •  Definition of the determinant of an endomorphism.
  •  Exterior algebra of a direct sum of modules.
  •  Twisted product of exterior algebras.
  •  Covariant extension of the exterior algebra of a module.
  •  Skew derivations on an exterior algebra.
  •  N-th exterior power of a matrix.
  •  Pfaffians.
  •  Symmetric algebra of a module (and of a free module).
  •  Symmetric algebra of a direct sum of modules.
  •  Covariant extension of the symmetric algebra of a module.
  •  Derivations on a symmetric algebra.
  •  Differential operators.
  •  Coalgebras.
  •  Graded coalgebras.
  •  Tensor product of coalgebras.
  •  Twisted tensor product of coalgebras.
  •  Commutative, skew-commutative coalgebras.
  •  The module of linear forms on a given coalgebra.
  •  Hopf algebras.
  •  Twisted Hopf algebras.
  •  Tensor product of Hopf algebras.
  •  Twisted product of twisted Hopf algebras.
  •  The exterior algebra of a module as a twisted Hopf algebra.
  •  The symmetric algebra of a module as a graded Hopf algebra.
  •  The Grassmann algebra of a module.
  •  Algebra of differential forms on the exterior algebra of a module.
  •  The graded dual of a graded module.
  •  The graded dual of an algebra.
  •  The graded dual of a coalgebra.
  •  The graded dual of a Hopf algebra.

 

 
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