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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.
- 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?
- Abstract Algebra GCD Proof Using Ideals [06/28/2005]
-
Can you prove that GCD(an + b, a(n+1) + b) = GCD(a, b)?
- 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]
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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]
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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...
- 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.
- Cubic Functions [5/13/1996]
-
Investigate the cubic functions of f(x) = ax^3+bx^2+cx+d...
- 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?
- 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?
- Equivalence Relations on Sets [2/3/1996]
-
Please tell me how many equivalence relations can be defined on the set S
= [a,b,c].
- Euclidean Domain [08/12/2003]
-
An integral domain with a division algorithm.
- Euclidean Domains and Quadratic Fields [08/12/2003]
-
How can I prove that there does not exist a division algorithm in any
quadratic field K = Q(sqrt(D)), where D <= -15?
- Extension Fields [12/03/1998]
-
Extension field proofs: show that Q(sqrt(2), sqrt(3)) = Q(sqrt(2) +
sqrt(3)). Find the splitting field of x^3 - 1 over Q.
- Factor Rings and Ideals [04/22/2003]
-
Give an example to show that a factor ring of an integral domain may
be a field. Show that R and R prime are isomorphic rings. Show that if
R has unity 1 and R prime has no 0 divisors, the phi (1) is unity for
R prime.
- Field Theory: Equal Sets [06/12/2002]
-
Show that Q(sqrt(i)) is isomorphic to Q(sqrt(2), i).
- Field Theory: Splitting Field [06/12/2002]
-
Find the splitting field of (x^3 - 5).
- Finding Integer Solutions of x^3 - y^2 = 2 [06/01/2000]
-
How can I find all integer solutions of the equation x^3 - y^2 = 2 and
prove that they are the only solutions?
- Finite Group: Prime Order Property [02/11/2003]
-
Suppose that G is a finite group with the property that every
nonidentity element has prime order. If Z(G) is not trivial, prove
that every nonidentity element of G has the same order.
- Finite Groups and Normal Subgroups [10/30/2004]
-
Let G be a finite group of order n such that G has a subgroup of order
d for every positive integer d dividing n. Prove that G has a proper
normal subgroup N such that G/N is Abelian.
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