Abstract Definition and 506 Threads

Could someone try to rank 'Abstract Algebra' textbooks, either undergraduate, or graduate level: By how rigorous they are, how they transfer to applicable subjects, and how well they're laid out, in a pedagogical manner. Any answers would be appreciated. Thanks in advance! SL!

Is the wave function describing reality or does it describe the observers uncertainty about the system? I say that it's real but I would like to hear any comments or evidence that suggest the wave function isn't a description that has a one to one correspondence with a underlying reality. It's...

Homework Statement Burglars are pushing, with a horizontal force Fpush, a safe of mass m and coefficient of kinetic friction μk up a slope of angle θ. What is the safe's acceleration (in abstract terms)? Homework Equations a[SUB]s= +/-gsinθ (natural accl down a slope) friction on a...

If \ast : (f \ast g)(n) = \sum\limits_{d|n}f(d)g(\frac{n}{d}), show that \ast is commutative. Note that d|n says d divides n. Now I was not sure how to do this from an abstract algebra point of view although when I stare at it my though process was to maybe rewrite it somehow, which will then be...

I am reading at the moment about abstract algebra. It is a very interesting field. I was amazed by the number of examples, applications and related concepts. Never seen something similar in any other mathematical field. I saw lots and lots of theorems and I was wondering whether I should...

I work retail, and spend hours each day folding clothes. I would like to ponder interesting facets of mathematics and logic, but am unable to (or plainly, do not). When I go to bed at night, I'd like to ponder abstract questions, yet do not. Basically, I see people (and read of people) who...

Homework Statement Define the set Q[√2] to be the set {a + b√2 | a, b are rationals}, and define addition and multiplication as "usual" (so 2×4 = 8, 2 + 4 = 6, you know, the usual). Show that for any nonzero A in the set Q[√2], there exists an inverse element so that A×A-1 = 1Q[√2]. There...

Homework Statement An interesting example of a ring: Begin with a nonempty set X and form the power set of X, P(X), which is the set of all subsets of X. On P(X), define addition + and multiplication × as follows: For A, B in P(X): A × B = A ∩ B A + B = (A\B) ∪ (B\A), where as...

I was always good at maths, just because primary/high school math was simple enough to find concrete examples for the abstract concepts, and that helped me a lot on exams. Since then I tried to grasp more advanced concepts. But I always faced with pure overformalized, overgeneralized stuff...

Hi there, Need one upper div math class to fill out my schedule. It looks like it's a choice between intro to abstract algebra or intro to topology. Which would benefit me more, as a student looking towards grad school?

Abstract Algebra: Relations; Find a relation that is symmetric, etc Homework Statement Find a relation that is symmetric and transitive but not reflexive. Homework Equations None, other than my chosen condition on the relation, namely: xy > |x + y|. The Attempt at a Solution...

Hello. I started Gallians Contemporary Abstract Algebra today. Is it wise to go through each of the given proofs for all of the theorems. For example I just studied the proof for division algorithm. Took Quite some time. I don't know if I could have produced this proof without peeking at the...

Hey everyone.. Not sure why I made this but I'm going to post it anyways haha. If you like to graphic design or have seen abstract art similar please post here:) I created this art from scratch. I attached the image.. created by using GIMP on a linux distro:)

If I were to use an abstract algebra book for quick and easy reference which one would it be? Dummit and Foote is very cumulative, is there anything better in the market? And how long would it take to work out all of D + F for an average student with basic background in Algebra?

Homework Statement Problem 35, Section 7.3 of Dummit and Foote: Let I, J, and K be ideals of R. (a) Prove that I(J+K) = IJ+IK and IJ+IK = I(J+K). (b) Prove that if J \subseteq I then I \cap (J + K) = J + (I \cap K). 2. Concern/Question Despite the problem statement specifically...

So I was trying to figure out a straightforward method to calculating the possible number of combinations on a beginner minesweeper game (81 squares, 9x9, 10 mines) I figure that i can attribute this to binary. Because the 9x9 part shouldn't really matter. It is essentially a 81 bit...

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...

Technological advances in the last several years (such as Japan's "dream machine") have given us crude glimpses into the visual component of the thought process. Is it possible to do the same with the auditory component of thoughts? Is there any scientific evidence suggesting that thought sounds...

well,I'm not quite sure is it appropiate to post this here. I'm just here for some help... I‘m a Chinese senior student in college. When I'm writing the abstract in my bachelor's thesis,I've got some trouble—— terminology and grammar. As you know,my English is not that good to write an english...

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...

Can unquantized fields be considered smooth curved abstract manifolds? Say free particle solutions of the Dirac equation or the Klein Gordon equation? Can quantized fields also be considered curved abstract manifolds? Thanks for any help!

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...

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...

Can anyone help me confirm if I've solved this correctly? Many thanks. Homework Statement Prove that \sqrt{ab}>\frac{2ab}{a+b} if a & b are positive & unequal. Homework Equations The Attempt at a Solution if (\sqrt{ab})^2>(\frac{2ab}{a+b})^2 if ab>\frac{4a^2b^2}{(a+b)^2} if...

Homework Statement The final answer I have of (a+b)(a-b) does not appear to fit the textbook's required "results of inequalities which hold true for all real no.s", i.e. either: 1. (a)^2 or (a-b)^2 or 2. -(a+b)^2. Can anyone confirm if I have solved this correctly, in line with the conditions...

Just an abstract question here. How different do you think the world would be if we were taught quantum mechanics before classical mechanics, given the prerequisite that we already have a good knowledge of the mathematics? Of course, it's a highly unlikely scenario, but an interesting one none...

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...

See attached picture. The question asks to prove that the statement which I have written on the first line is true. But I somehow proceeded to proving it is false. Basically what I did was simplify the given expression into the form (P or Q) => R and said this is equivalent to (P=>R) ^...

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...

Homework Statement problem didn't state, but I assume let V be a vector space: V = C^3 and scalar is C Homework Equations Define a non-zero linear functional T on C^3 such that T ((1, 1, 1)) = T ((1, 1, −1)) = 0 The Attempt at a Solution So let X1 = (1, 1, 1); X2 = (1, 1, -1); It...

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...