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Browse College Modern Algebra

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Subgroups of the Rational Numbers Under Addition [02/01/2003]
I need to describe all the subgroups of the rational numbers under addition.

Sylow P-Subgroups of Symmetric Groups [05/13/2009]
Let p be an odd prime. First, find a set of generators for a p-Sylow subgroup K of S_p^2 (the symmetric group with degree p^2). Then find the order of K and determine whether it is normal in S_p^2 and if it is Abelian.

Symbol for Irrational Numbers? [09/23/2002]
What is the standard symbol used to represent the irrational numbers? Is it Q-bar?

Symmetries of a Cube [10/09/2003]
Prove that the group of symmetries of a cube is isomorphic to S_4.

Tensors and Spinors Defined [09/30/1999]
What are tensors and spinors? Can you explain giving examples?

Two Integers and a Third Degree Polynomial: Square in Z? A Galois Theory Proof [07/28/2010]
A student seeks to prove that there exist infinitely many pairs of non-zero integers such that a particular third degree polynomial is square in the ring of integers. Since the exercise appears in a chapter on Galois theory, Doctor Jacques expands the scope of the question to proving that there are infinitely many such polynomials.

Uniqueness of Ideals [09/23/2003]
How can I prove that in Mn(Q) (the ring of n*n matrices over the rational numbers Q), (0) and Mn(Q) are the only ideals?

Using Galois theory to prove that x^4 +1 is reducible in Z_p[X] for all primes p [11/09/2008]
A student sees a Dr. Math proof that x^4 + 1 is reducible in Z_p[X] for all primes p, but seeks an alternate method -- one using Galois theory.

What is a Torsion Subgroup? [03/05/2003]
Let G be an Abelian group. Show that the elements of finite order in G form a subgroup. This subgroup is called the torsion subgroup of G. Now find the torsion subgroup of the multiplicative group R* of nonzero real numbers.

What is Knot Theory? [03/10/1998]
Researching how knot theory is related to topology and modern algebra.

What Makes Polynomials Relatively Prime? [11/20/2007]
Why are polynomials whose only common factors are constants considered 'relatively prime'? Why are the common constants not considered? For example, 3x + 6 and 3x^2 + 12 are considered relatively prime even though they have a common constant factor of 3.

Zero-Factor Theorem [02/08/2002]
If a and b are real numbers, and if ab = 0, then a = 0 or b = 0. Why the restriction 'a and b are real numbers'?

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