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Browse College Modern Algebra

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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?

3-Dimensional 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 sub-groups.

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 Swinnerton-Dyer Conjecture [06/20/2012]
What's the Birch and Swinnerton-Dyer Conjecture? Doctor Vogler outlines how "B and SD" connects the rank of elliptic curves with their Hasse-Weil L-functions.

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 n-dimensional 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 (p-1) divides (n-1) 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