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Browse College Modern Algebra

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An Integral Domain, Closed in its Field of Fractions [01/20/2012]
An adult teaching himself field theory wonders if Z[2^(1/3)] is integrally closed in Q[2^ (1/3)]. By invoking Dedekind's Criterion and considering minimal polynomials, Doctor Jacques steps through a proof, then offers a less robust but simpler method.

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)?

Inverses in the Field GF(2^8) in AES [03/29/2010]
A programmer needs to compute inverses of polynomials that have hexadecimal coefficients other than {00} and {01}. Doctor Vogler helps by clearing up the notation that appears in the Federal Information Processing Standard (FIPS) Advanced Encryption Standard (AES).

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...

Is There a 'Discriminant' for a Quartic Equation? [01/12/2005]
Is there a way to determine the nature of the roots of a quartic equation in the form ax^4 + bx^3 + cx^2 + dx = 0 by simply using the coefficients, as with the discriminant b^2 - 4ac in a quadratic equation of the form ax^2 + bx + c = 0?

Is There a "Discriminant" for a Quartic Equation ... in Closed Form? [02/17/2012]
A modern algebra student picks up the thread from another student's earlier conversation with Doctor Vogler. Together, they re-visit and lay the question to rest, applying Sturm's Theorem in the process.

Klein Four Group and Isomorphism Proof [11/01/2004]
Let |G| = 4. Prove that either G is isomorphic to C4, or G is isomorphic to V. What is the group V(Klein four group)?

Lagrange's Theorem [01/24/2002]
Let G be a finite group of order p, where p is a prime number and G is a cyclic group. I need the proof of the theorem.

Lagrangian Notation [04/08/1999]
Using Lagrange's Theorem to calculate the index of a subgroup.

Let k Be a Field [04/20/1999]
Prove or disprove that a prime ideal I of the ring k[x] is a maximal ideal...

Linear Independence of Square Roots of Primes [11/07/1996]
How do you prove that the square roots of a finite set of different primes are linearly independent over the field of rationals?

The Many Binary Operations of a Two Element Set [11/03/2011]
A student struggles to conceive of all the binary operations possible in a two-element set. Doctor Peterson clarifies the scope of the abstraction before enumerating pairs and offering a template for organizing them.

Mathematical Deduction [07/22/1997]
Prove: Let E be a subset of P such that i) 1 is in E; ii) whenever n is in E, also n+1 is in E. Then E = P.

Matrix Algebra [08/28/1997]
I am not sure which formula of matrices to use in this situation.

Matrix Multiplication, Finite Fields [07/13/2001]
What is matrix multiplication over the Galois field GF(2^8)?

Measure Theory and Sigma Algebras [03/24/2003]
I'm trying to understand what a 'measure' is.

Modern Algebra [07/10/1997]
Show that the natural log of i^1/2 = i times pi over 4.

Modern Algebra [07/23/1997]
Let n be a positive integer, and define f(n)= 1!+2!+3!+...+n!. Find polynomials P(x) and Q(x) such that f(n+2)=P(n)f(n+1)+Q(n)f(n) for all n > or = 1.

Modern Algebra Proof [01/29/2001]
If G is a finite group whose order is even, show that G contains an element of order two.

Monstrous Moonshine Conjecture [11/12/1998]
I've been reading about the monstrous moonshine conjecture. Can you explain more on the j function and the Monster Group?

Multiplicative Groups of Order (p-1) [05/12/2000]
What is the proof that primitive roots for multiplicative groups of order (p-1), where p is prime, exist? Is there an algorithm for finding them?

Newton Sums and Monic Polynomial Roots [11/06/2004]
There are 311 distinct solutions to the equation x^311 = 311x + 311. These solutions are designated by the 311 variables a_1,a_2,....a_311. Find the sum (a_1)^311 + (a_2)^311 + (a_3)^311 + ... + (a_311)^311. I've been told that Newton Sums can be used on this problem, but I'm not sure how to apply it. Can you help?

Noether Rings [12/01/1997]
What are Noether rings and how do they work?

Noether's Rings [08/20/1999]
Can you explain what Noetherian rings are, and a little of the math behind them?

Non-Abelian Groups [02/11/2003]
Given a group in which every a satisfies a^3 = 1, is that group abelian?

Only Two Abelian Groups [02/25/2003]
Show that any group with order p^2, p is a prime, is Abelian. Show that up to isomorphism that only two such groups exist.

Operator-Version of Schroeder-Bernstein [11/13/1997]
My question relates to some algebraic structures, grupoids...

The Order of an Element [11/05/1998]
Suppose that G is a group that has exactly one nontrivial proper subgroup. Prove that G is cyclic and |G|=p^2, where p is prime...

Permutation Groups Generated by 3-Cycles [05/14/2003]
Show A_n contains every 3-cycle if n >= 3; show A_n is generated by 3- cycles for n >= 3; let r and s be fixed elements of {1, 2,..., n} for n >= 3 and show that A_n is generated by the n 'special' 3-cycles of the form (r, s, i) for 1 <= i <= n.

Plotting Complex Numbers [07/22/1997]
I cannot figure out (1-i)^2i = 2^ie^1.570796.

Polynomial Congruence [02/28/2001]
Find a polynomial (F) in Field(7) with degree less then 4...

Polynomial Proof [03/26/2001]
Can I prove that if p(x) is a polynomial of nth degree with integer coefficients in x, then p(a) = b, p(b) = c, and p(c) = a?

Polynomials of the Fifth Degree and Above [07/28/2001]
I know how to find the root of a polynomial of the form: ax^2+bx+c=0. But what about a polynomial of the third degree?

Primitive Elements vs. Generators [05/24/2002]
Prove that x is a primitive element modulo 97 where x is not congruent to 0 if and only if x^32 and x^48 are not congruent to 1 (mod 97).

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