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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.

The Algebra that High School and College Share in Common [12/05/2016]
In modern algebra, a teen does not recognize any of the algebra that he learned in high school. Doctor Vogler emphasizes operations to explain the connection.

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 only if the Galois group of f is a 2-group.

Construction of a Regular Heptadecagon [12/27/2009]
Gauss derived a finite algebraic expression for sin(pi/17) which led to an algorithm for the construction of the regular 17-gon. Can you help me understand the derivation of his expression?

Counting Solutions of Quadratic Diophantine Equations [02/21/2009]
Doctor Vogler helps a computer scientist enter the ring of Gaussian integers as they delve into the three categories of integer solutions to x^2 + y^2 = 2*N^2.

Cubic Functions [5/13/1996]
Investigate the cubic functions of f(x) = ax^3+bx^2+cx+d...

Cyclic Group Proof [12/15/2008]
How can I prove that if a group G has a unique subgroup of order d for every d that divides |G|, then G is cyclic?

Cyclic Groups [04/18/2002]
Prove that a group of order 5 is cyclic.

Cyclic Groups [02/27/2003]
Prove that the group of nonzero rational numbers under multiplication is not cyclic.

Cyclic Groups [03/10/2003]
Prove that an infinite group must have an infinite number of subgroups.

Cyclic Groups [06/25/2003]
I am supposed to prove that every subgroup of a cyclic group is characteristic.

Cyclic Subgroups: Finite Groups [02/01/2002]
Is there a noncyclic subgroup of order 4 in U(40)? If so how can it be found?

Defining (|R)^n in a Field [03/27/2001]
What multiplication operation would define (|R)^n in a field?

Discussion of Euclidean Functions of Z [06/10/2008]
Can you help me give a description of all Euclidean functions of Z? The common example is of course the absolute value function, but it seems to me that other weird Euclidean functions can be constructed, too.

D is Not Euclidean [02/20/2003]
Let a be a negative integer. Show that Z[a^0.5] is a Euclidean domain if and only if a = -1 or a = -2.

Drawing Regular N-gons (Compass and Straightedge) [11/17/1997]
Is it true that the only regular n-gons that can be drawn using ONLY a straightedge and compass are those with the number of sides equal to a Fermat Prime or a product of Fermat Primes?

Elliptic Curves: Algorithms [03/11/1999]
Find the number of points on the curve over F sub p for an elliptic curve y^2 = x^3 + 1.

Epimorphism Proof [1/2/1998]
What is a proof that, in the category of groups, an epimorphism is just an onto homomorphism?

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