Ask Dr. Math

College Archive

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]

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 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.

Page:  1  2  3  4 [next>]