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. M

    Abstract Algebra: Is It Too Difficult for Calculus?

    Currently I am reviewing basic algebra, trigonometry and I will also be starting calculus this fall semester... I enjoy reading about math and I wanted to know what abstract algebra is? Would this be to difficult to read seeing that I am only starting calculus? If so what other types of...
  2. BloodyFrozen

    Number Theory, Linear & Abstract Algebra

    Are there any basic prerequisites before learning about these branches of mathematics?
  3. N

    [abstract algebra] is this ring isomorphic to

    Homework Statement Consider \frac{\mathbb Z_2[X]}{X^2+1}, is this ring isomorphic to \mathbb Z_2 \oplus \mathbb Z_2, \mathbb Z_4 or \mathbb F_4 or to none of these? Homework Equations / The Attempt at a Solution - \mathbb F_4 No, because \mathbb Z_2[X] is a principle ideal domain...
  4. B

    Modern Algebra & Real Analysis: Learn Proofwriting?

    Hello, I just took ordinary diff eq and I've had calc III and linear algebra, but I'm worried about taking Modern Algebra or Real Analysis next semester because I have no experience writing proofs. The linear algebra class was all computation on tests and homework (we did see some proofs on...
  5. I

    What is the identity element in abstract algebra groups?

    The .pdf can be ignored. Let A + B = (A - B) U (B - A) also known as the symmetric difference. 1. Look for the identity and let e be the identity element A + e = A (A - e) U (e - A) = A Now there are two cases: 1. (A - e) = A This equation can be interpreted as removing from A all elements...
  6. Z

    Courses Which should I take: Abstract Algebra vs 4th year lab (non-thesis) course?

    I am undecided between these two for 2012 spring term (my last semester hopefully)
  7. P

    Does Every Group of Prime Power Order Have a Subgroup of Prime Order?

    Homework Statement Let G be a group with pk elements, where p is a prime number and k is greater than or equal to 1. Prove that G has a subgroup of order p. The Attempt at a Solution I attempted to prove this by showing that the conditions for a set to be subgroup form a subgroup of order p...
  8. T

    Is Re-taking Abstract Algebra Necessary for a Strong Foundation in Mathematics?

    I really feel dissapointed in myself that I didn't perform as well as I wanted last semester. I took Modern Algebra I and Geometry. The Geometry class covered Euclidean and non-Euclidean geometries. I bombed the final but earned an overall of a B+ because of a 90-something percentile homework...
  9. MathWarrior

    Where is abstract algebra used?

    I've been studying cryptography and I found out that AES uses Galois Fields. I was therefore wondering where else does abstract algebra pop-up for real world use?
  10. F

    Grad textbook on abstract algebra

    What is the absolute best abstract algebra book for graduate students? I was wanting a book that covers algebra in the most comprehensive manner possible, at about the level of Hungerford's Algebra. I was wondering if Carstensen's Abstract Algebra in the Sigma Series in Pure Mathematics is a...
  11. M

    Prime ideal question (abstract algebra)

    Homework Statement Let D = Z[sqrt(10)], and let P be the ideal (2,sqrt(10)) 10). Prove that P is a prime ideal of D. Homework Equations The Attempt at a Solution Not sure where to start. I think elements are of the for a+b*sqrt(10), a,b integers. Any hints as to what to do next?
  12. D

    Is Abstract Algebra the Key to Unlocking Mathematical Concepts?

    I have started to write Abstract Algebra notes as I am learning them, and typing them with LaTex afterwards. I have just done a bit but I want some of you to help and see if I have got any thing wrong (having the wrong concept in your mind can have terrible consequences) or anything else to make...
  13. R

    Abstract algebra, left and right translations, what are these good for?

    Homework Statement Does anyone know what left and right translations are good for? \begin{cases} R_{a}g=ga\\L_{a}g=ag\end{cases} with a,g\in G and G is a Lie group How can we interpret these relations in the easiest way like we try to explain it to a student which if not familiar with...
  14. N

    Exploring Abstract Algebra: Helpful Links and Practice Problems

    I was wondering if anyone knew any links on the Internet that help to explain abstract algebra and maybe works through some problems as well. Thank you in advance
  15. V

    Abstract Algebra - Polynomials: Irreducibles and Unique Factorization

    Homework Statement Show that x^2\,+\,x can be factored in two ways in \mathbb{Z}_6[x] as the product of nonconstant polynomials that are not units.Homework Equations Theorem 4.8 Let R be an integral domain. then f(x) is a unit in R[x] if and only if f(x) is a constant polynomial that is a...
  16. M

    Proving |H intersect K| = q for subgroup H and K in G of order pqr.

    Homework Statement Suppose |G| = pqr where p, q, and r are distinct primes. If H is a subgroup of G and K is a subgroup of G with |H| = pq and |K| = qr, then |H intersect K| = q. Homework Equations NA The Attempt at a Solution I have so far: Let a be an element of H intersect K...
  17. N

    Introduction to Abstract Algebra

    I was wondering if anyone could give me any links or an introduction to abstract algebra. I know that abstract algebra is a tough concept to understand (at least for some people, but it varies from person to person). If anyone could help with the basics of it would be greatly appreciated.
  18. M

    Simple Abstract Algebra Problem

    Homework Statement Let G be a nonempty finite set closed under an associative operation such that both the left and right cancellation laws hold. Show that G under this operation is a group. Homework Equations My book defines the left and right cancellation laws as : "For any a,b in...
  19. H

    Abstract Algebra: Proving G is Isomorphic to H with Log

    1. Homework Statement [/b] The set of positive real numbers, R+, is a group under normal multiplication. The set of real numbers, R, is a group under normal addition. For the sake of clarity, we'll call these groups G and H respectively. Prove that G is isomorphic to H under the isomorphism...
  20. D

    Abstract Algebra Hello Experts: Proving Theorems About Ideals and Radicals

    Hello Experts, I can't find the proof of this theorems please help me: Given that there is a commutative ring R and 2 ideals I and J, also given that I is included in J I need to prove 1) radical of I is in radical of J 2) radical of radical of ideal I = radical of ideal I...
  21. A

    How do mathematicians think about abstract algebra?

    Hi Folks. I was hoping to pick the brains of some of the mathematicians and mathematically inclined on this site. I'm very interested in how mathematicians think about abstract objects that don't seem to be grounded in anything concrete. In particular, how do mathematicians think to...
  22. B

    Abstract Algebra and cyclic subgroups

    Homework Statement from Algebra by Michael Artin, chapter 2, question 5 under section 2(subgroups) An nth root of unity is a complex number z such that z^n =1. Prove that the nth roots of unity form a cyclic subgroup of C^(x) (the complex numbers under multiplication) of order n...
  23. MathWarrior

    Abstract Algebra vs Number Theory?

    I was wondering if one wanted to pursue learning more about cryptography which of these classes would be the most important? Number theory of abstract algebra?
  24. L

    Abstract Algebra Question: Maximal Ideals

    Homework Statement a) Show that there is exactly one maximal ideal in Z_8 and in Z_9. b) Show that Z_10 and Z_15 have more than one maximal ideal. Homework Equations I know a maximal ideal is one that is not contained within any other ideal (except for the ring itself) By...
  25. S

    Abstract Algebra: Properties of the Group U(n)

    Homework Statement (This is an example of a group in my text). An integer 'a' has a multiplicative inverse modulo n iff 'a' and 'n' are relatively prime. So for each n > 1, we define U(n) to be the set of all positive integers less than 'n' and relatively prime to 'n'. Then U(n) is a group...
  26. S

    Abstract Algebra: Question About the Elements in U(n)

    Homework Statement For any integer n>2, show that there are at least two elements in U(n) that satisfy x^2 = 1. Homework Equations None The Attempt at a Solution If the definition of the group U(n) is "the set of all positive integers less than n and relatively prime to n" then the...
  27. S

    Applications of abstract algebra to engineering

    I was wondering if there are any applications of abstract algebra to engineering and where I can go to learn about them?
  28. M

    Abstract Algebra: define an operation

    Homework Statement Does the rule g*x = xg^-1 define an operation of G on G? Homework Equations The Attempt at a Solution I don't even know what this means. Could someone just tell me what it means for a rule to define an operation of one group on itself? I should be able to figure...
  29. K

    Do I need a lot of abstract algebra knowledge to start learning Lie algebra

    I'm a physics undergrad and doing some undergrad study on QFT, and I found that Lie algebra is often invoked in texts, so I decide to take a Lie algebra this sem but I've not taken any abstract algebra course before.The first day's class really beats me because the lecturer used many concepts...
  30. C

    Abstract algebra questions relating to Ideals and cardinality of factor rings

    Homework Statement Find the number of elements in the ring Z_5[x]/I, where I is a) the ideal generated by x^4+4, and b) where I is the ideal generated by x^4+4 and x^2+3x+1. Homework Equations Can't think of any. The Attempt at a Solution I started by finding the zeros of the...
  31. N

    Is Abstract Algebra Worth Taking for Physics?

    Would this be a good thing to take? More specifically, will a introduction to this shed light on/put on more solid ground many of the techniques/organizations in physics that are presented as "tricks"? I just want to be sure it will be worth it, since i'll be taking it alongside...
  32. K

    Proving Normality of Subgroups in Cyclic Groups

    I'm taking this course "abstract algebra" at university and I've been given some homework questions. I was able to solve all of them but one. And it would be great if anyone could help me with this. The question is like this: "If all cyclic subgroups of G are normal, then show that all...
  33. D

    Abstract Algebra: Groups of Permutations

    Homework Statement List the elements of the cyclic subgroup of S_6 generated by f = \left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6\\ 2 & 3 & 4 & 1 & 6 & 5\\ \end{array}\right)Homework Equations The Attempt at a Solution I really do not understand what the elements of a permutation really...
  34. M

    Should I take Combinatorics or Abstract Algebra?

    I am a senior student double majoring in computer science and mathematics with the intention of getting a p.h.d in theoretical computer science(either computational complexity or applied discrete mathematics). for the upcoming winter semester I can take 1 math course. The ones that are related...
  35. B

    Determinant proof from abstract algebra

    Homework Statement Let A be a a square n*n matrix. Prove that A^-1 has only integer enteries if and only if the determinant of A is + or -1. Homework Equations general knowledge of determinants The Attempt at a Solution Proof: => Suppose that det(A) = 1 (without losing...
  36. C

    Abstract Algebra First Isomorphsm Theorem

    Homework Statement Use the First Isomorphism Theorem to show that Q[x]/(x^3-3) is isomorphic to {a+b*sqrt(3)} Homework Equations First Isomorphism Theorem: If f: G-> H is a homomorphism then G/ker(f) is isomorphic to im(f) The Attempt at a Solution I understand that I need to show...
  37. T

    Solve Abstract Algebra Homomorphism Problems with Step-by-Step Guidance

    abstract algebra ...HELP, PLZ! THIS IS THE PROBLEM: COMPUTE THE INDICATED QUANTITIES FOR THE GIVEN HOMOMORPHISM KER (PHI) AND PHI(18) FOR PHI: Z -> Z10 (SUBCRIPT) SUCH THAT PHI(1)=6 Can anyone please help me to solve this problem? I don't even know what it's asking for? Don't know where...
  38. Truecrimson

    The best edition of Fraleigh's Abstract Algebra?

    I'm going to buy A First Course in Abstract Algebra by Fraleigh. I've looked at 6th and 7th ed. 6th doesn't have a section on homology groups, but 7th does. From what I found from other threads here, 4th also has homology groups, and 3rd is at least good on group actions. (I haven't got to group...
  39. B

    Abstract Algebra, Division Ring question

    Homework Statement Let R = { [ a + b*sqrt(m) c + d*sqrt(m) ] } [ n(c - d*sqrt(m)) a - b*sqrt(m) ] (Sorry if the matrix is unclear... I can't get it space nicely. r11 = a + b*sqrt(m) r12 = c + d*sqrt(m) r21 = n(c -d*sqrt(m))...
  40. S

    Solve Abstract Algebra Homework: Finding Left Coset (1,2,3)H with Permutations

    One of my homework problems asks me to list the left coset (1,2,3)H where σ=(1,4,5)(2,3) and H=<σ>. I know that you have to take the do the permutation of (1,2,3)(1,4,5)(2,3) but i am not sure how you can do that? I got (1,2,3)H={(1,2,3)(3)(1,2,4,5)} but i do not think that is right
  41. J

    Abstract Algebra, rings, zero divisors, and cartesian product

    The problem states: Let R and S be nonzero rings. Show that R x S contains zero divisors. I had to look up what a nonzero ring was. This means the ring contains at least one nonzero element. R x S is the Cartesian Product so if we have two rings R and S If r1 r2 belong to R and s1 s1...
  42. C

    Abstract algebra-> Let R be a ring and let M2(R) be the set of 2 x 2 matrices with

    Abstract algebra--> Let R be a ring and let M2(R) be the set of 2 x 2 matrices with Homework Statement Let R be a ring and let M2(R) be the set of 2 x 2 matrices with entries in R. De fine a function f by: f(r) = (r 0) <----matrix ...(0 r) for any r ∈ R (a) Show that f is a...
  43. E

    Abstract Algebra: conjugates of cyclical groups

    Homework Statement If G is a group with operation * and \alpha,\beta\in G, then \beta\ast\alpha\ast\beta^{-1} is called a conjugate of G. Compute the number of conjugates of each 3-cycle in S_{n} (n\geq3). Homework Equations The Attempt at a Solution For any group S_{n} there...
  44. N

    How Does Cycle Length Relate to Element Order in Group Representations?

    abstract algebra question?? here is the problem from abstract algebra, anyone could help? Thanks a lot! let G be a finite group. Show that in the disjoint cycle form of the right regular representation Tg(x)=xg of G each cycle has length | g |. (Tg(x) means T sub g of x) loofinf...
  45. silvermane

    Proving Greatest Common Divisor of a,b,c | Abstract Algebra 1

    1. The problem statement: Consider 3 positive integers, a, b, c. Let d_{1} = gcd(b,c) = 1. Prove that the greatest number dividing all three of a, b, c is gcd(d_{1},c) 3. My go at the proof and thoughts: Well, I know that the common divisors of a and b are precisely the divisors of...
  46. A

    Abstract algebra vector space problem

    Homework Statement We have a vector space (V, R, +, *) (R being Real numbers, sorry I couldn't get latex work..) with basis V = span( v1,v2). We also have bijection f: R² -> V, such as f(x,y) = x*v1+y*v2. Assume you have inner-product ( . , . ): V x V -> R. ( you can use it abstractly and...
  47. C

    Abstract algebra proof involving prime numbers

    The question states prove, If p is prime and p | a^n then p^n | a^n I am pretty sure I have i just may need someone to help clean it up. There are two relevant theorems i have for this. the first says p is prime if and if p has the property that if p | ab then p | a or p | b the...
  48. G

    Abstract Algebra- A simple problem with Cosets

    I need to find all the cosets of the subgroup H={ [0], [4], [8] ,[12] } in the group Z_16 and find the index of [Z16 : H]. Help would be appreciated :)
  49. O

    Abstract Algebra: Homomorphism f Determined by f(1) in Z

    Homework Statement Let R be any ring and f:Z→R a homomorphism. a)Show that f is completely determined by the single value f(1) b)Determine all possible homomorphisms f in the case when R = Z. Homework Equations The Attempt at a Solution This question has me totally confused...
  50. E

    Abstract Algebra- homomorphisms and Isomorphisms, proving not cyclic

    1. Suppose that H and K are distinct subgroups of G of index 2. Prove that H intersect K is a normal subgroup of G of index 4 and that G/(H intersect K) is not cyclic. 2. Homework Equations - the back of my book says to use the Second Isomorphism Theorem for the first part which is... If K...
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