Status Updates From A Course in Algebra

??. Vinberg 5.56-5.58

Ch5 vector space!

Exercise 4.99. The sign of a cyclic permutation is sign(i1,i2,...,ip) = (-1)^(p-1)

In a Monoid, the left e is the right e, and a left inverse is also the right one. Refer to Lang.

Theorem 4.67 (Lagrange's Theorem). Let G be a finite group and H its
subgroup. Then
\G\ = \G:H\\H\.

A subgroup of GL_n(R) called the orthogonal group, is denoted O_n.

Group of permutations or the symmetric group on n elements is denoted S_n.

1. Nonzero elements of a field K form an abelian group with respect to multiplication. It is called the multiplicative group of K and is denoted K*.
2. If A is an associative ring with unity, the set of its invertible elements is a multiplicative group as well. We denote this group as A*.

Group GL_n(K) is isomorphic to the group GL(K^n)

The general linear group of V is denoted GL(V).
Nonsingular square matrices of order n over a field K form a multiplicative group denoted GL_n(K).

Ch 4 Elements of Group Theory

The quotient field of the ring K[x] of polynomials over a field K is called the field of rational fractions (or rational functions) over the field K and is denoted K(x).

Quotient field (field of fractions) of the ring A is denoted Q(A).

Cubic resolution of quartic function, refer to http://en.wikipedia.org/wiki/Quartic_... and http://en.wikipedia.org/wiki/Klein_fo...

Ch3 is now within reach

Let A be an integral domain. Elements a and b are called associated (notation a ~ b) if b | a and a | b.

In the first four chapters I tried to make the presentation sufficiently detailed to be suitable for ... a mathematics freshman ... In later chapters I allowed myself to skip details .... -- so the real trip hasn't begun yet

Wilson's Theorem: (p-1)!=-1(mod p), p is a prime.

the algebra K[[x]] has no zero divisors

Exercise2.69.For a linear map \phi: F(X,K) -> F(Y,K), where X be the set of tetrahedron's edges and Y the set of its faces, find dim Ker \phi when char K=2.

Remark 1.83. Observe that all constructions in the last three sections (1.7 Vector spaces, 1.8 Algebras, 1.9 Matrix Algebra) would remain unchanged if we replaced K with a commutative associative ring with unity, for instance, the ring of integers or a ring of residue classes. The only difference lies in terminology: in this more general situation the term module is used instead of vector space (see Section 9.3).

Theorem 1.49. The ring Zn is a field if and only if n is a prime number.