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College Archive

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?

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]

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...

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 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?

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.

Factoring Quartic Expressions with No Real Zeros [07/19/2009]

I know that x^4 + 7x^2 - 2x + 15 = (x^2 - x + 3)(x^2 + x + 5) because I got the quartic by multiplying the quadratics. But if I were simply given the quartic to factor, how how would I do it? I'd try to find real zeros but there aren't any and then I would be stuck.

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.

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