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Browse College Modern Algebra

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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?

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?

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).

Producing MOD(x,y) with Arithmetic Operations [03/04/1998]

Is there any way to produce the "MOD(dividend,divisor)" spreadsheet function using basic arithmetic operations?

Product of a Finite Abelian Group [06/28/2004]

Suppose G = {a1, a2, ... , an} is a finite Abelian group. If G has odd order, what can you say about the 'product,' a1*a2*...*an, of all the elements of G? What can you say about this 'product' if G has even order? What if G is not Abelian?

Product of Disjoint Cycles [10/16/1998]

How to express (1 2 3 5 7)(2 4 7 6) as the product of disjoint cycles.

Proof of Division Algorithm [11/13/1997]

a,b are positive integars, b does not equal 0; there are unique integers q and r such that a = qb+r; 0 is less than or equal to r, r is less than modulus value of b.

Proof of Normal Sylow p-Subgroup [10/28/2005]

Let G be a finite group in which (ab)^p = a^p*b^p for every a,b in G, where p is a prime dividing O(G). Prove that the Sylow p-subgroup of G is normal.

Proof of One Step Subgroup Test [05/13/2002]

Prove that a nonempty subset H of a group G is a subgroup of G if and only if a*b^(-1) is in H for all a, b in H.

Proof of Only One Identity Properity for Binary Operations [10/31/2001]

I am trying to prove that there is one and only one identity property for every operation.

Proof of Subgroup Involving an Isomorphism [11/04/2004]

Suppose ö is an isomorphism from a group G to a group G'. Prove that if K is a subgroup of G, then ö(K) = {ö(k) | k is in K} is a subgroup of G'.

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