Abstract algebra Definition and 459 Threads

It's been a while since I've posted. This is a problem I had for a homework assignment a few weeks ago but I completely figure out. Any help appreciated. Homework Statement "A card-shuffling machine always rearranges cards in the same way relative to the order in which they were given to...

I need to choose one more math class to reach a full-time status for next fall. So far I am already taking Classical Mech I from Physics Dept, Analysis I and PDE from Math Dept. I hear Analysis is already time-consuming hard class and I guess PDE isn't easy either, so I am considering to...

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...

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

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...

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.

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...

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...

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...

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?

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...

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...

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...

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...

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...

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...

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) =...

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...

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...

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.

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...

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}...

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...

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)...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

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...

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.

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...

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?

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

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...

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

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...

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...

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...

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...

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...

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...