What is Abstract algebra: Definition and 457 Discussions

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra.
Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures.
Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called variety of groups.

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  1. N

    Abstract Algebra: Non-trivial Rings Containing Only Zero-Divisors

    Homework Statement Is there a finite non-trivial ring such that for some a, b in R, ac = bc for all c in R? Does there exist finite non-trivial rings all of whose elements are zero-divisors or zero? 2. The attempt at a solution Let a, b ≠ 0 in R such that ac=bc for all c in R...
  2. J

    Proving this basic fact about the annihilator in abstract algebra

    Maybe I'm misinterpreting the question, I'm not sure how to prove that n_0 i = 0.
  3. C

    Abstract Algebra- Finding the Minimal Polynomial

    Homework Statement Given field extension C of Q, Find the minimal polynomial of a=sqrt( 5 + sqrt(23) ) (element of C).Homework Equations The Attempt at a Solution I may be complicating things, but let me know if you see something missing. Doing the appropriate algebra, I manipulated the above...
  4. C

    Abstract Algebra- Conjugate Problem

    Homework Statement Let G be a group of odd order, and a an element of G (not identity). Show that a and a^-1 are not conugate. Homework Equations The Attempt at a Solution The only hint I have is to consider action of G on itself by conjugation.
  5. D

    Where can I find helpful resources for abstract algebra?

    I was wondering if anyone has compiled a list of AA resources. Recently, I have found that I practically need to learn everything from the class outside of class all over again. I have been playing around with YouTube, but haven't really found anything worthwhile. So, what about you guys...
  6. N

    Tricky abstract algebra problem

    Homework Statement Prove that SL_{2}(ℝ) is generated by the set: [1 a], [1 0] [0 1], [b 1], a,b \in ℝ Homework Equations GCD (Greatest common divisor) The property of special linear group Some basic linear algebra, like determinant The Attempt at a Solution SL_{2}(ℝ) is the group...
  7. C

    [Abstract Algebra] Maximal Ideal

    Homework Statement Let F be the field and f(x)=x-1,g(x)=x^2-1 and F[x]/(f(x)) is isomorphism to F, is it g(x) maximal?? 2. The attempt at a solution I will say no.Since g(x) is not 0, the dieal (x^2-1) in a prime idea domain F is maximal iff (x^2-1) is irreducible. And we say...
  8. C

    Unit in a ring (abstract algebra)

    Homework Statement Is (x^2-1) a unit in F[x]? where F is a field. 2. The attempt at a solution I might say yes, cause we can find the taylor expansion of 1/(x^2-1), is my idea right?
  9. L

    Abstract Algebra during final year?

    Good morning everyone. So I've been thinking quite a bit about it and recently switched from applied math to pure math, and I wish to attend grad school, if not PhD then at least a master's with thesis. I'm in the middle of my 2nd year, so next Fall I plan on taking Analysis, and then the fall...
  10. K

    Is it normal to be so discouraged by abstract algebra?

    I'm currently in my first abstract algebra course, focused on sets, groups, arithmetic modulo, rings, fields etc. I've never taken an abstract course before. I've taken: Pre-calc Calc 1-2 Linear Algebra Advanced Applied Linear Algebra so the concept of abstraction is very new to me; I...
  11. K

    How Does Subset Proof in Abstract Algebra Work?

    Homework Statement Question 1. Let U be a universal set, A and B two subsets of U. (1) Show that B ⊆ A ∪ (B ∩ A^c). (2) A = B if and only if there exists a subset X of U such that A ∪ X = B ∪ X and X\A^c = X\B^c. The Attempt at a Solution My attempt at a solution is as follows...
  12. A

    Abstract Algebra - Natural Numbers Proof

    The question is which sets of natural numbers are closed under addition. I know that odd is not, and I know how to prove that sets of multiples are, but my professor said there is something more and that is has to do with greatest common divisor. He said to pick numbers like 3 and 5 or 5 and 8...
  13. micromass

    Algebra Abstract Algebra by Dummit and Foote

    Author: David Dummit, Richard Foote Title: Abstract Algebra Amazon link https://www.amazon.com/dp/0471433349/?tag=pfamazon01-20 Prerequisities: Being acquainted with proofs and rigorous mathematics. Level: Undergrad Table of Contents: Preface Preliminaries Basics Properties of the...
  14. micromass

    Algebra A book of Abstract Algebra by Pinter

    Author: Charles Pinter Title: A book of Abstract Algebra Amazon link https://www.amazon.com/dp/0486474178/?tag=pfamazon01-20 Prerequisities: High-school algebra Level: Undergrad Table of Contents: Preface Why Abstract Algebra? History of Algebra New Algebras Algebraic Structures...
  15. B

    Abstract Algebra Proof Using the First Isomorphism Theory

    Homework Statement See attatchment. I couldn't upload the picture. 2. The attempt at a solution I have the following: Define mapping f: ℝ2 -> ℝ as follows: f(x,y) = 3x - 4y Claim: f is a homomorphism Pick any (x,y) in ℝ2. Then f(x,y) = f(x)*f(y) = 3x - 4y = (x+x+x)-(y+y+y+y) =...
  16. T

    A mapping from an integral domain to non-negative integers, Abstract Algebra

    So just had this question as extra credit on a final: Let D be an integral domain, and suppose f is a non-constant map from D to the non-negative integers, with f(xy) = f(x)f(y). Show that if a has an inverse in D, f(a) = 1. Couldn't figure it out in time. I was thinking the way to go...
  17. S

    Abstract Algebra - Group of Order 12 with Conjugacy Class of Order 4

    Homework Statement A group G of order 12 contains a conjugacy class C(x) of order 4. Prove that the center of G is trivial.Homework Equations |G| = |Z(x)| * |C(x)| (Z(x) is the centralizer of an element x\inG, the center of a group will be denoted as Z(G)) The Attempt at a Solution Let G...
  18. S

    Abstract Algebra: Finite Field

    Show that every finite field with p+1 elements, where p is a prime number, is commutative. I know this has something to do with composite numbers, but I'm not quite sure how to show this.
  19. S

    Abstract Algebra: Rings, Unit Elements, Fields

    1) Show that (R,*,+) is a ring, where (x*y)=x+y+2 and (x+y)=2xy+4x+4y+6. Find the set of unit elements for the second operation. I understand that the Ring Axioms is 1. (R,+) is an albein group. 2. Multiplication is associative and 3. Multiplication distributes. I just don't understand how to...
  20. A

    Abstract Algebra homework Direct products

    Homework Statement We've shown if G_{1},G_{2},...,G_{n} are subgroups of G s.t. 1)G_{1},G_{2},...,G_{n} are all normal 2)Every element of G can be written as g_{1}g_{2}...g_{n} with g_{i}\inG 3)For 1\leqi\leqn, G_{i}\capG_{1},G_{2},...,G_{i-1}=e then G\congG_{1}xG_{2}x...xG_{n}...
  21. N

    Abstract Algebra HW: Show nk=kn for N,K ∈ G

    Homework Statement Suppose N \lhd G and K \vartriangleleft G and N \cap K = \{e\}. Show that if n \in N and k \in K, then nk = kn. Hint: nk = kn if and only if nkn^{-1}k^{-1} = e. Homework Equations These "relevant equations" were not provided with the problem I'm just putting them here to...
  22. F

    Abstract Algebra Order of Permutation

    Homework Statement See image. Homework Equations The Attempt at a Solution I am finding the orders of permutations. I know that you first find the orbits or cycles I don't know the difference (but I should). This is what my professor said: If you have (1345)(897)...
  23. H

    What Are the Proofs for Powers in Normal Subgroups and Orders in Homomorphisms?

    Homework Statement a) Let H be a normal subgroup of G. If the index of H in G is n, show that y^n \in H for all y \in G. b) Let \varphi : G \rightarrow G' be a homomorphism and suppose that x \in G has order n. Prove that the order of \varphi(x) (in the group G') divides n. (Suggestion: Use...
  24. Z

    Abstract Algebra, order of ab is equal to the order of a times the order of b?

    Abstract Algebra, order of ab is equal to the order of a times the order of b?? Hi! I am working on some problems in abstract algebra and I am stuck at the moment. I hope some of you guys could help me out a little. Homework Statement a and b are two elements in a group G. Assume that...
  25. R

    Abstract algebra, finite A-module

    Homework Statement Let A be an integral domain with field of fractions K, and suppose that f\in A is non zero and not a unit. Prove that A[\frac{1}{f}] is not a finite A-module. [Hint: if it has a finite set of generators then prove that 1,f^{-1},f^{-2},...,f^{-k} is a set of generators for...
  26. U

    Unique Decomposition of Elements in an Abelian Group

    Homework Statement Let A be an abelian group, written additively, and let n be a positive integer such that nx=0 for all x \in A. Such an integer n is called an exponent for A. Assume that we can write n=rs, where r, s are positive relatively prime integers. Let A_{r} consist of all x \in A...
  27. G

    Testing Screwed up Abstract Algebra exam unsure if I have the ability to do math.

    After getting back a result in an Abstract Algebra exam (In which I only got 70%), a result just below the class average I am having extreme doubts about my ability to become a mathematician. The real shock was that I believed I understood the material well enough to get at least 90%. I am...
  28. H

    Abstract Algebra: repeating decimals and prime factors

    Homework Statement Prove if m/n has a repeating decimal expansion of period k, and n has no repeated prime factors, then some prime factor of n divides 10k-1 and no number of the form 10j-1 for 1 ≤ j < k Homework Equations The Attempt at a Solution I know that if a decimal...
  29. srfriggen

    Abstract Algebra: List elements of Subgroup

    Homework Statement List the elements of the subgroups <3> and <7> in U(20). Homework Equations The Attempt at a Solution U(20)= {1, 3, 7, 9, 11, 13, 17, 19} = <3> = <7>. So basically I have that the common elements of, <3> and <7> and U(20), under + modulo 20, are all...
  30. R

    Unique factorization domain, roots of a polynomial, abstract algebra

    Homework Statement let A be a UFD and K its field of fractions. and f\in A[x] where f(x)=x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0} is a monic polynomial. Prove that if f has a root \alpha=\frac{c}{d}\in K,K=Frac(A) then in fact \alpha\in A I need some guidance with the proof. Proof...
  31. srfriggen

    Abstract Algebra, Group Question

    Homework Statement (a) Suppose a belongs to a group and lal=5. Prove that C(a)=C(a3). (b) Find an element a from some group such that lal=6 and C(a)≠C(a3). Homework Equations The Attempt at a Solution For (a) I know I need to show that every element in the set C(a) is...
  32. srfriggen

    Abstract Algebra question regarding coprimes

    Homework Statement For any integer n>2, show that there are at least two elements in U(n) that satisfy x2=1. Homework Equations In my book I found a definition: Define U(n) to be the set of all positive integers less than n and relatively prime to n. The Attempt at a Solution...
  33. P

    Early Abstract Algebra Problem - Pinter's Textbook

    Homework Statement This problem is from Charles C. Pinter's A Book of Abstract Algebra, Second Edition. The problem is B7 of Chapter 2.Show that the operation * is either associative or not. x*y=\frac{xy}{x+y+1} This problem seems simple to me: I keep arriving at YES for an answer...
  34. M

    How Can Someone Learn Abstract Algebra form Harvard ?

    Hello , I've a question . How Can someone Get Harvard Books on Mathematics generally and Abstract Algebra Specially ? How Can The one Buy it ?
  35. K

    Abstract Algebra vs Linear Algebra

    Hey guys, As of now, I am in a sets and logic proof based course (Intro to proof-writing). This course basically teaches logic, how to write proofs using examples of algebraic equations, sets: power sets, unions and intersections of classes, etc. With a C in this course, you can register...
  36. P

    Taking Topology, Real Analysis and Abstract Algebra concurrently a good idea?

    Hello all, In the Fall I am planning on taking Real Analysis, Abstract Algebra and doing an independent study in something(my professor has yet to get back to me on what he is willing to do it in). My question is would it be too much of a workload to try and do another independent study in...
  37. B

    Abstract algebra or ODE for physics

    currently i am a math major, still unsure whether pure or applied. i am also looking to double major in physics. which class would be more helpful to me: abstract algebra, or (upper division) ODE class? I have taken the lower division DE class already.
  38. J

    A Question about Pinter's A Book of Abstract Algebra

    Hello all, I am currently doing a self-study in Abstract Algebra. I was a math major in college (not so long ago), so I have some exposure to upper level math. For one reason or another, I wanted to go back and re-learn Abstract. I was using Fraleigh until I discovered Pinter's text which...
  39. K

    Which Abstract Algebra Textbook is Best for Self-Study?

    I need to buy a textbook for self study in abstract algebra for self study. Although I'm a physics major, I have lot's of experience with proof. I'm between Artin's Algebra and Dummit's Abstract Algebra. Which one do you recommend?
  40. R

    Difficulty of abstract algebra in relation to calculus

    How difficult is abstract algebra or group theory, plus complex analysis in relation to calculus?
  41. R

    Abstract algebra: proving an ideal is maximal, Constructing quotient rings

    Homework Statement M = {(pa,b) | a, b are integers and p is prime} Prove that M is a maximal ideal in Z x Z Homework Equations The Attempt at a Solution I know that there are two ways to prove an ideal is maximal: You can show that, in the ring R, whenever J is an ideal such...
  42. D

    Is this the perfect outfit for an Abstract Algebra class?

    https://www.amazon.com/dp/1111569622/?tag=pfamazon01-20 Any idea?
  43. M

    Result for f(a+b+c) = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc

    Homework Statement f is a quadratic function from the second degree and f(a)=bc;f(b)=ac;f(c)=ab Homework Equations Calculate : f(a+b+c) The Attempt at a Solution Can we say that f(a+b+c)=f(a)+f(b)+f(c) and the go on from there plugging in the values of each one are do i have to do...
  44. M

    Help Develop Tongue in Cheek Abstract Algebra Proof

    Help Develop "Tongue in Cheek" Abstract Algebra Proof Hello all, First and foremost I would like to thank everyone on the forum. Your post here have been invaluable in aiding me in completing many of my engineering courses. Now I am attempting to develop a "tongue in cheek" proof using...
  45. R

    Abstract algebra: monic gcd of polynomials in a subfield problem

    Homework Statement Let K \subseteq L be fields. Let f, g \in K[x] and h a gcd of f and g in L[x]. To show: if h is monic then h \in K[x]. The Attempt at a Solution Assume h is monic. Know that: h = xf + yg for some x, y \in K[x]. So the ideal generated by h, (h) in L[x] equals...
  46. T

    Abstract Algebra: isomorphism proof

    Homework Statement Let G be an abelian group of order n. Define phi: G --> G by phi(a) = a^m, where a is in G. Prove that if gcd(m,n) = 1 then phi is an isomorphism Homework Equations phi(a) = a^m, where a is in G gcd(m,n) = 1 The Attempt at a Solution I know since G is an...
  47. J

    Abstract Algebra mathematica add on

    hey, I have this group I've been trying to generate using the GenerateGroupoidByRelations[] function but it keeps giving me an error, G = GenerateGroupoidByRelations[{a, b}, {a^4 == e, b^4 == e, a ** b ** a ** b == e, a^3 ** b ** a^3 ** b == e}, SizeLimit -> 60] gives...
  48. A

    Is 2Z isomorphic to 4Z? (Abstract algebra)

    Actually I'm stupid today, it happens once in a while that I get extremely lazy and stupid in mathematics, but today I came up with a bizarre thing in abstract algebra that I couldn't find my mistake on my own and I'm not sure whether what I've concluded is true or wrong, I was proving another...
  49. L

    Understanding Group Size Change: G/N in Abstract Algebra

    This is not really a homework questions, rather a concept based one. I am studying from Fraleigh's ''Intro to abstract algebra'' and in chapter 15 it states, that for a group G and normal non-trivial subgroup of N of G, the factor group G/N will be smaller than G. I am not sure how he counts the...
  50. Shackleford

    Understanding Left Cosets in Abstract Algebra

    http://i111.photobucket.com/albums/n149/camarolt4z28/untitled.jpg G/N is the set of all left cosets of N in G. I don't understand the notation. a) The permutations are (1,2), (2,3), (3,1). What are the left cosets - <1>, <2>, <3>? That doesn't make sense with permutations. b) I have...
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