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Browse College Modern Algebra

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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 Dimensional Quotient Space [11/01/2008]

If V is a finite dimensional vector space and w is a subspace of V then prove that the quotient space v/w is also finite dimensional.

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.

Finite Group Theory--Sylow Subgroups [07/09/2004]

Let U and W be subsets of a Sylow p-subgroup P (of a group G) such that U and W are normal in P. Show that U is conjugate to W in G iff U is conjugate to W is the normalizer of P.

Finite Simple Groups [3/15/1996]

What is the Monster - large finite simple group - and why is it interesting?

Fnding the ln of a Negative Number [07/22/1997]

Can you give me a general formula for this problem? Like ln(-3)?

Four Modern Algebra Problems [07/22/1997]

Could you please get me started on these homework problems?

Galois Groups [04/28/2005]

Suppose K > Q (Q = rational numbers) is an extension of degree 4 which is not a Galois extension. Let L be the Galois closure of K over Q. Prove: 1. The galois group Aut_Q(L) is isomorphic to S4,A4, or D8 (dihedral group of order 8). 2. Aut_Q(L) is isomorphic to D8 if and only if K contains a quadratic extension of Q.

Galois Theory [06/27/1997]

Does factorization of polynomials have anything in common with Galois Theory? What are Galois groups?

Galois Theory and Cyclic Extensions [04/28/2005]

Let a be an element of the algebraic closure of the rational numbers (Q{bar}), but not in the rational numbers, and let F be a subfield of Q{bar} that is maximal for the property that a is not in F. Prove that every finite extension of F is cyclic.

Galois Theory and Finite Permutation Groups [06/01/2007]

Do the elements (123456)(78) and (17) generate the symmetric group of order eight?

Galois Theory Proofs [02/22/2003]

Prove that a sum of two algebraic numbers is also an algebraic number.

Galois Theory/Splitting Fields [03/07/2002]

Determine the splitting field of the polynomial x^p - x - a over F_p where a is not equal to zero and a is an element of F_p. Show explicitly that the Galois group is cyclic.

Generators for the Group S_n [10/08/2001]

Show that S_n, the symmetric group on n elements, is generated by the elements: (1 2) and (1 2 3 ... n) for all n greater than or equal to 2.

Group Proof [02/22/2002]

Prove that if G is a group of order p^2 (p is prime), and G is not cyclic, then a^p = e for each a in G.

Groups and Subgroups [10/20/2001]

Show that a subset of a group is a subgroup. Understanding how properties distinguish elements.

Groups, Subgroups, Cosets, and Order [04/20/2000]

Show that every element of the quotient group G = Q/Z has finite order; prove that a group of order n has a proper subgroup if and only if n is composite; suppose H, K contained in G are subgroups of orders 5 and 8, respectively and prove that H intersects K = {e}.

Group Theory [02/08/2002]

Show that if G is a group s.t., (a*b)^i = a^i*b^i for three consecutive integers i for all a,b in G, then G must be abelian.

Group theory [11/22/1994]

The four rotational symmetries of the square satisfy the four requirements for a group, and so they are called a subgroup of the full symmetry group. (Notice that the identity is one of these rotational symmetries and that the product of two rotations is another rotation in the subgroup.) a. Do the four line symmetries of the square form a subgroup? b. Does the symmetry group of the equilateral triangle have a subgroup?

Herstein Commutivity Problem [09/20/2002]

Prove that if x = x^3 for all x in R (a ring), then R is commutative.

How Does the Dot Product of Two Vectors Work? [03/17/2006]

I'm confused about the dot product. How does multiplying two vectors give you the angle they form together?

How Do You Make a Character Table? [10/15/2008]

I'm taking a course in Representation Theory and am having trouble with character tables. Can you explain the ideas behind constructing them?

Ideals of Z [07/30/2003]

What are the ideals of Z? Find the prime ideals of Z. What are the maximal ideals of Z?

Idempotents of Z(n) [10/10/2000]

What are the idempotents of Z(n) when n is twice a prime?

Identity Element [02/09/2002]

Let f be an element of S_n. Show that there exists a positive integer k such that f^k = i, the identity function.

The Importance of 1 Not Being a Prime Number [04/25/2009]

Does division by zero make sense in a field of cardinality 1? If there is only one element, the multiplicative and additive identities must be the same, so 0 = 1 and 1/0 = 1/1 = 1. Is that true?

Inconstructible Regular Polygon [02/22/2002]

I've been trying to find a proof that a regular polygon with n sides is inconstructible if n is not a Fermat prime number.

Integer Proof Using Diophantine Equation [01/10/2005]

How do you prove that the integer 26 is the only integer preceded by a a square (25) and followed by a cube (27)?

Integer Rings [05/18/2005]

Given a (commutative, characteristic zero) field F, is (are) there any ring(s) such that F is the fraction field of R? If any, how many, and how are they related?

Integrals of Rational Functions by Partial Fractions [12/20/2007]

Can you explain finding integrals of rational functions using partial fractions?

Intersection of Normal Subgroups [03/18/1999]

Hints for proving that the intersection of two normal subgroups is a normal subgroup.

Inverse of an Inverse [01/20/2002]

In a group, prove that (a^-1)^-1 = a for all a.

Inverse Quaternions [12/01/1999]

How do you calculate inverse quaternions? For example, the inverse of 3 - 4i + 5j + 6k.

Inverses in the Field GF(2^8) [11/07/2000]

How can I get the multiplicative inverse of a byte in the polynomial field GF(2^8)?

Irrational Powers [8/30/1996]

Does an irrational number to the irrational power yield a rational number?

Isomorphic Groups [02/11/2003]

Is the additive group of rationals isomorphic to the multiplicative group of non-zero rationals?

Isomorphic Groups and Subrings [04/15/1998]

I have a few problems on isomorphic groups and subrings that I just can't figure out...

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