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 TOPICS This page:   modern algebra    Search   Dr. Math See also the Internet Library:   modern algebra COLLEGE Algorithms Analysis Algebra    linear algebra    modern algebra Calculus Definitions Discrete Math Exponents Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean Imaginary/Complex   Numbers Logic/Set Theory Number Theory Physics Probability Statistics Trigonometry Browse College Modern Algebra Stars indicate particularly interesting answers or good places to begin browsing. 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). 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? Page: []

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