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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. 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). Page: []

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