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Browse College Modern Algebra
Stars indicate particularly interesting answers or
good places to begin browsing.
 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 nonzero 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.
 ZeroFactor 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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