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Browse College Modern Algebra
Stars indicate particularly interesting answers or
good places to begin browsing.
 Galois Theory [11/20/1996]

Please explain Galois Theory.
 An Introduction to Groups in Abstract Math [04/23/2008]

Can you recommend how to start learning about abstract math in general
and groups in particular?
 3Dimensional Rotation Space [05/18/2009]

Consider a closed loop representing a rotation of 2pi in RP^3. Can you
show that one cannot continuously deform this loop to a point?
 Abelian Groups [05/15/2000]

How do I prove that the operation @, defined by a@b = a^ln(b), is an
abelian group for the set of positive real numbers not equal to 1?
 Abelian Groups [09/28/2001]

Let G be a group with the following property: If a, b and c belong to G
and ab = ca, then b = c. Prove that G is Abelian.
 Abelian Groups [10/22/2003]

Let G be a group with the identity element e. Show that:
1) if x^2 = e for all x in G, then G is Abelian;
2) if (xy)^2 = x^2 * y^2 for all x,y in G, then G is Abelian.
 Abelian Groups [09/14/2005]

If a and b are any elements of a group G and (ab)^3 = a^3*b^3, is G
necessarily Abelian?
 Abelian Groups Cyclic [03/05/2002]

Prove that every abelian group of order 6 is cyclic.
 Abelian Group Tables [04/29/1999]

How do you construct the first Abelian group for the general case?
 About Finite Groups [02/03/2003]

If H is a nonempty subset of the finite group {G,*} with the property
that x*y is in H when x and y are in H, is H a subgroup of G?
 About Symmetry About a Line [06/06/2012]

A modern algebra student seeks symmetry in the table of a commutative operation.
Doctor Peterson clarifies what it means to exhibit symmetry about a line.
 Abstract Algebra GCD Proof Using Ideals [06/28/2005]

Can you prove that GCD(an + b, a(n+1) + b) = GCD(a, b)?
 An Algebraically Closed Field, Its Multiplicative Group, and Its Isomorphic Subgroup [02/05/2012]

Given an algebraically closed field, a student wonders about the equivalence of its
multiplicative group and an isomorphic subgroup. Doctor Vogler provides two
counterexamples of injective mappings that are not surjective.
 Algebraic Extensions [06/28/1997]

What are algebraic extensions?
 Algebraic Structures [02/22/1999]

Two questions on subgroups.
 Automorphism of a Finite Group [11/02/2004]

If some automorphism T sends more than three quarters of elements into
their inverses, prove that T(x) = x^(1) for all x in G, where G is
finite.
 Automorphism on a Finite Group [10/12/2001]

Let G be a finite group, f an automorphism of G such that f^2 is the
identity automorphism of G. Suppose that f(x)=x implies that x=e (the
identity). Prove that G is abelian and f(a)=a^1 for all a in G.
 Beginning Modern Algebra Proofs [02/02/1999]

Let Nm be the set of natural numbers < m. Prove that for any m>2, there
exists k in Nm that is not a perfect square mod m...
 The Birch and SwinnertonDyer Conjecture [06/20/2012]

What's the Birch and SwinnertonDyer Conjecture? Doctor Vogler outlines how "B and
SD" connects the rank of elliptic curves with their HasseWeil Lfunctions.
 Can A Negative Integer Be Factored Into Primes? [11/11/2003]

Can the number 103,845 have the prime factors of 3, 5, 7, 23, and 43?
We find this confusing because we have been told a positive number can
have prime factors but a negative number can't.
 Cardinality of Euclidean Space [09/13/2005]

What is the cardinal number of a ndimensional Euclidean space R^n
where n tends to aleph_0, aleph_1, aleph_2, and so on?
 Carmichael Numbers [10/31/1997]

Why must a Carmichael number be the product of at least three distinct
primes? Why is n a Carmichael number iff (p1) divides (n1) for every
prime p dividing n?
 Commutative Ring, Maximal Ideal [12/08/2003]

Prove that in a comutative ring any ideal is contained in some maximal
ideal.
 Constructibility and Galois Groups [04/30/2005]

Let a be a complex number and a root of an irreducible polynomial f
over the rationals. Show that a is constructible if and onl
