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Browse College Modern Algebra

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Proof That G Is Abelian [03/05/2003]

Let H be a subgroup of G that is different from G and let x*y=y*x for all x and y in G minus H.

Prove G is a Cyclic Group [02/27/2003]

Let group G be finite Abelian such that G has the property that for each positive integer n the set {x in G such that x^n = identity} has at most n elements. Prove G is a cyclic group.

Prove Sylow-p Subgroups Abelian [05/15/2003]

G is a finite simple group with exact 2p + 1 sylow-p subgroups. Prove that each of these sylow-p subgroups is Abelian.

Proving an Element is in a Group [04/12/2002]

If G is a group, prove that the only element g in G with g^2 = g is 1.

Proving That Z_{mn} is Isomorphic to Z_m X Z_n [04/22/2009]

If m and n are relatively prime, show that Zmn is isomorphic to Zm X Zn.

Quadratic Diophantine Equation [01/16/2009]

Find all positive integers N such that 2*N^2 - 2*N + 1 is the square of an odd integer.

Quadratic Fields and the Division Algorithm [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?

Quaternary Numbers [6/28/1995]

How do quaternary numbers work? As I understand it, its q = r + a.i + b.j + c.k where r is real and i^2 = j^2 = k^2 = -1, but what happens when you start multiplying and dividing i, j and k?

Quaternion Numbers [01/23/1997]

How do you divide quaternion numbers? Can quaternion math be extended to transcendental functions?

Quaternion Numbers in Quantum Physics [04/11/2001]

Are there any applications of quaternion numbers in quantum physics?

Questions in Modern Algebra [2/3/1996]

I am studying Modern Algebra (grad level course) and would appreciate your help in answering three questions...

Rainbow Logic [1/13/1995]

In the Family Math book, there is an activity called Rainbow Logic. It's like battleship in that one player sets up a matrix and the other player(s) have to match their matrices by asking questions...

Real Numbers Closed under Division [09/24/2002]

Is it true that real numbers are not closed under division because we can't divide by 0?

Resultant of Two Polynomials [09/13/2001]

Can you give me an example of how to find the resultant of two polynomials?

Rings and Ideals [11/13/1998]

Can you help me with the following proofs on rings, ideals, and polynomials? If R is a commutative ring, prove R[x]/(x) = R. ...

Ring Theory - a Telescoping Problem [01/20/2002]

An element a in a ring R is said to be nilpotent if there exists a positive integer n such that a^n = 0. Show that if a is nilpotent, a-1 is a unit.

Roots of Cubic Equations [7/9/1995]

How many real roots does a third degree equation in the form ax^3 + bx^2 + cx + d = 0 have, and what are they?

Roots of the Cubic Equation in F2^M [07/20/2000]

Is there a general solution for the cubic equation where x is an element of the finite field F2^M?

Show a Subset [02/09/2003]

If the number of elements in a finite group G with identity e is even...

Show That G is a Group [02/28/2002]

Let G be a finite group. Show that there exists a positive integer "m" such that a^m = e for all a in G. Suppose that G is a set closed under an associative operation such that: for every a,y in G, there exists an x in G such that ax = y; and for every a,w in G, there exsits a u in G such that ua = w. Show that G is a group.

Significance of Rational Numbers [01/11/2003]

Why are rational numbers defined the way they are?

Solvable Groups [12/10/2002]

What is the connection between the solvability of polynomials for degree>=5 and the solvability of An for n>=5? What is the proof that shows An is simple for n>=5?

Solving Nonlinear Systems with Many Variables [10/25/2000]

Do you know a numeric algorithm that solves systems of non-linear polynomial equations with up to 10 variables?

Special Unitary Groups In Physics [11/09/1998]

Could you explain the groups SU(2) and SU(3)? They are central to descriptions of quantum chromodynamics.

Splitting Fields of Quartic Polynomials [05/14/2004]

I picked the irreducible polynomial x^4 - 8x^2 + 8 and I tried to find its splitting field (E) and G(E/Q) set, but I got stuck after finding the roots. Could you please help me out? Thanks in advance!

Subgroup and Order of a Group [11/06/2003]

If o(G) = p^n, p is a prime number, and H is not equal to G and it is a subgroup of G, show that there exists an x that is an element of G and x is not an element of H such that x^(-1)Hx = H.

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?

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?

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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