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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. Euclidean Domain [08/12/2003] An integral domain with a division algorithm. Euclidean Domains and Quadratic Fields [08/12/2003] How can I prove that there does not exist a division algorithm in any quadratic field K = Q(sqrt(D)), where D <= -15? Extension Fields [12/03/1998] Extension field proofs: show that Q(sqrt(2), sqrt(3)) = Q(sqrt(2) + sqrt(3)). Find the splitting field of x^3 - 1 over Q. Factoring Quartic Expressions with No Real Zeros [07/19/2009] I know that x^4 + 7x^2 - 2x + 15 = (x^2 - x + 3)(x^2 + x + 5) because I got the quartic by multiplying the quadratics. But if I were simply given the quartic to factor, how how would I do it? I'd try to find real zeros but there aren't any and then I would be stuck. Factor Rings and Ideals [04/22/2003] Give an example to show that a factor ring of an integral domain may be a field. Show that R and R prime are isomorphic rings. Show that if R has unity 1 and R prime has no 0 divisors, the phi (1) is unity for R prime. Fields, and Adding and Multiplying in Them [08/13/2015] A teen seeks to prove that no proper subset of the field of rational numbers is itself a field. Doctor Vogler suggests a way forward, and further clarifies the scope of addition and multiplication in modern algebra. 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 Galois Groups and Subgroups of a Splitting Field of Degree 8 [03/29/2010] Using minimal polynomials, a geometric representation, and some intuition, Doctor Jacques works through the Galois groups and subgroups of a splitting field of a polynomial over Q which has sqrt(2 + sqrt(6)) as one root. 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? 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. Page: []

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