Abstract algebra Definition and 459 Threads

So I have two sets, call it ##A## and ##B##. I also have a function ##f:A\rightarrow B##. By themselves, it does not matter (or at the very least make sense) to think of ##A## and ##B## as, say, groups (I'm not really thinking exclusively about groups, just as an example). For that matter, it...

What are good books in universal algebra, given that I have a background in Herstein (Topics in Algebra), Hubbard/Hubbard, Engelking (Topology), and Dugundji (Topology)? I am currently reading Hungerford, and I found a field called universal algebra while searching internet for some concepts...

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What is the most motivating way to introduce group theory to first year undergraduate students? I am looking for some real life motivation or something which has a real impact.

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Homework Statement I am supposed to prove or disporve that ##f:\mathbb{R} \rightarrow \mathbb{R}## ##f(x)=\sqrt{x}## is onto. And prove or disprove that it is one to one Homework EquationsThe Attempt at a Solution I know for certain that this function is not onto given the codomain of real...

Hi. I have the following question: Let G be a finite group. Let K be a subgroup of G and let N be a normal subgroup of G. Let P be a Sylow p-subgroup of K. Is PN/N is a Sylow p-subgroup of KN/N? Here is what I think. Since PN/N \cong P/(P \cap N), then PN/N is a p-subgroup of KN/N. Now...

Homework Statement Show that the group (\mathbb Z_4,_{+4}) is isomorphic to (\langle i\rangle,\cdot)? Homework Equations -Group isomorphism The Attempt at a Solution Let \mathbb Z_4=\{0,1,2,3\}. (\mathbb Z_4,_{+4}) can be represented using Cayley's table: \begin{array}{c|lcr} {_{+4}} & 0 &...

Homework Statement Suppose X is a set with n elements. Prove that Bij(X) ≅ S_n. Homework Equations S_n = Symmetric set ≅ = isomorphism Definition: Let G and G2 be groups. G and G2 are called Isomorphic if there exists a bijection ϑ:G->G2 such that for all x,y∈G, ϑ(xy) = ϑ(x)ϑ(y) where the...

Homework Statement For each ##n \in \mathbb{N}##, let ##A_{n}=\left\{n\right\}##. What are ##\bigcup_{n\in\mathbb{N}}A_{n}## and ##\bigcap_{n\in\mathbb{N}}A_{n}##. Homework Equations The Attempt at a Solution I know that this involves natural numbers some how, I am just confused on a...

Homework Statement If gcd(f(x),g(x)) = 1 and m,n ∈ ℕ, show that gcd(f(x)^m, g(x)^n) = 1. Homework EquationsThe Attempt at a Solution So I had previously proved this for non-polynomials: gcd(a,b)=1 then gcd(a^n,b^n)=1 Proof: a = p1*p2*...*pn b = p1*p2*...*pm then a^n = p1^n*p2^n*...*pn^n...

Homework Statement Suppose a field F has n elements and F=(a_1,a_2,...,a_n). Show that the polynomial w(x)=(x-a_1)(x-a_2)...(x-a_n)+1_F has no roots in F, where 1_f denotes the multiplicative identity in F. Homework EquationsThe Attempt at a Solution Strategy: We have this polynomial...

Homework Statement 1. Let g(x) = x^4+46. a) Factor g(x) completely in ℚ[x]. b) Factor g(x) completely in ℝ[x]. c) Factor g(x) completely in ℂ[x]. 2. Completely factor the given polynomial in ℤ_5. [4]_5 x^3 + [2]_5 x^2 + x + [3]_5 Homework Equations ℚ = {m/n / m and n belong to Z, m is not...

Homework Statement Let F = {0,1,α,α+1}. Find all irreducible polynomials over F of degree at most 2. Homework EquationsThe Attempt at a Solution To determine an irreducible polynomial over F, I think it is sufficient to check the polynomial whether has a root(s) in F, So far, I got...

Homework Statement Let g(x) ∈ ℤ[x] have degree at least 2, and let p be a prime number such that: (i) the leading coefficient of g(x) is not divisible by p. (ii) every other coefficient of g(x) is divisible by p. (iii) the constant term of g(x) is not divisible by p^2. a) Show that if a ∈ ℤ...

Homework Statement Find all real numbers k such that x^2+kx+k is reducible in ℝ[x]. Homework EquationsThe Attempt at a Solution This seems like it is simple, but it is new to me so I am looking for confirmation. We know we can find the roots of a polynomial with b^2-4ab. We want b^2-4ab to be...

Homework Statement Let T:R-> S be a homomorphism of rings. Show that T(0_r) = 0_s. Homework EquationsThe Attempt at a Solution First off, the terminology used is kinda confusing. I take 0_r to be the zero in R. Is this correct? For some reason I recall my teacher quickly saying that it was...

This is a question that came about while I attempting to prove that a simple extension was a splitting field via mutual containment. This isn't actually the problem, however, it seems like the argument I'm using shouldn't be exclusive to my problem. Here is my attempt at convincing myself that...

Homework Statement Let R be a ring and suppose r ∈R such that r^2 = 0. Show that (1+r) has a multiplicative inverse in R. Homework Equations A multiplicative inverse if (1+r)*x = 1 where x is some element in R. The Attempt at a Solution We know we have to use two facts. 1. Multiplicative...

Homework Statement Suppose R is a commutative ring with only a finite number of elements and no zero divisors. Show that R is a field. Homework Equations Unit is an element in R which has a multiplicative inverse. If s∈R with r*s = 1. A zero divisor is an element r∈R such that there exists...

I'm taking an abstract algebra course that uses Hungerford's "An Introduction to Abstract Algebra" 3rd Ed. And while I feel like I'm following the material sufficiently and can do most of the proofs it's hard to learn and practice the material without a solutions guide. How am I supposed to know...

Homework Statement Consider the ring of polynomails in two variables over a field K: R=K[x,y] a)Show the elements x and y are relatively prime b) Show that it is not possible to write 1=p(x,y)x+q(x,y)y with p,q \in R c) Show R is not a principle ideal domain Homework Equations None The...

Hi everyone. So it's apparent that G/N cyclic --> G cyclic. But the converse does not seem to hold; in fact, from what I can discern, given N cyclic, all we need for G/N cyclic is that G is finitely generated. That is, if G=<g1,...,gn>, we can construct: G/N=<(g1 * ... *gn)*k> Where k is the...

Homework Statement Hello all I apologize for the triviality of this: Im new to this stuff (its easy but unfamiliar) I was wondering if someone could verify this: Find the G.C.D of a= 14+2i and b=21+26i . a,b \in \mathbb{Z} [ i ] - Gaussian Integers Homework Equations None The Attempt...

Homework Statement Good day, I need to show that S_n=\mathbb{Z}_2(semi direct product)Alt(n) Where S_n is the symmetric group and Alt(n) is the alternating group (group of even permutations) note: I do not know the latex code for semi direct product Homework Equations none The Attempt at...

Homework Statement Good day all! (p.s I don't know why every time I type latex [ tex ] ... [ / tex ] a new line is started..sorry for this being so "spread" out) So I was wondering if my understanding of this is correct: The Question asks: "\mathbb{Z}_4 has a subgroup is isomorphic to...

Homework Statement Good day, I need to show: \mathbb{Z}_{4}\oplus \mathbb{Z}_{4} is not isomorphic to \mathbb{Z}_{4}\oplus \mathbb{Z}_{2}\oplus \mathbb{Z}_{2} Homework Equations None The Attempt at a Solution I was given the hint that to look at the elements of order 4 in a group. I know...

Homework Statement Good day all, Im completely stumped on how to show this: |AN|=(|A||N|/A intersect N|) Here: A and N are subgroups in G and N is a normal subgroup. I denote the order on N by |N| Homework Equations [/B] Second Isomorphism TheoremThe Attempt at a Solution Well, I know...

I have a question about Automorphisms. Please check the following statement for validity... An automorphism of a group should map generators to generators. Suppose it didn't, well then the group structure wouldn't be preserved and since automorphisms are homomorphisms this would be a...

Homework Statement Coproducts exist in Grp. This starts on page 71. of his Algebra. Homework Equations [/B] Allow me to present the proof in it's entirety, modified only where it's convenient or necessary for TeXing it. I've underlined areas where I have issues and bold bracketed off my...

Dear Physics Forum advisers, Today, I got two gifts from my research mentor: "Topics in Algebra" by I.N. Herstein and "Abstract Algebra" by Dummit/Foote. I am very happy and grateful for his gifts, but I already have been studying the abstract algebra through Michael Artin and Hoffman/Kunze...

A couple of notes first: 1. \hom_{A}(-,N) is the left-exact functor I'm referring to; Lang gives an exercise in the section preceeding to show this. 2. This might be my own idiosyncrasy but I write TFDC to mean 'The following diagram commutes' 3. Titles are short, so I know that the hom-functor...

Dear Physics Forum advisers, I am a rising college junior in U.S. with a major in mathematics, and an aspiring applied mathematician. I apologize for this sudden interruption, but I wrote this email to seek your advice on my current problem on the course selection. I will very soon be...

This isn't homework, I'm just trying to refresh my memory on cyclic groups. My question is, in this problem solution, how does ##{\sigma_i}^m=1## follow from ##\sigma_i## being disjoint?

Homework Statement Hello guys So I have the following problem, given the mapping above I have to check weather it's ring homomorphism, and maybe monomorphism or epimorphism. The Attempt at a Solution So the mapping is obviously well defined, and I have proven it's homomorphism, and it's...

Dear Physics Forum advisers, My name is Phoenix, a sophomore with major in mathematics and an aspiring applied mathematician in the theoretical computing. I wrote this email to seek your recommendation on the textbooks for abstract algebra. I want to self-study the abstract algebra during...

Homework Statement Let F be a finite field of characteristic p∈{2,3,5}. Consider the quaternionic ring, Q_F={a_1+a_i i+a_j j+a_k k|a_1,a_i,a_j,a_k ∈ F}. Prove that Q_F is not a division ring. Homework EquationsThe Attempt at a Solution Let α=1+i,β=1+i+j∈QF. Then...

Homework Statement Show √ 2 + √ 3 algebraic over Q. Find its degree over Q. Prove the answer. Homework EquationsThe Attempt at a Solution Let ##\alpha= \sqrt{2}+\sqrt{3}\in \mathbb{R}##, then ##\alpha^4-10\alpha^2+1=0## which is a root of ##f(x)=x^4-10x^2+1## where ##f(x)## in...

Homework Statement Construct $\mathbb{F}_{16}$ as a quotient of $\mathbb{Z}_2[X]$. How many non-zero elements are primitive in this field? Calculate $|GL2_(\mathbb{F}_16)|$. Homework Equations Primitive Theorem The Attempt at a Solution For the first question, I don't know how to construct...

Dear Physics Forum advisers, I am a college sophomore in US with a major in mathematics, and an aspiring algebraic number theorist and cryptographer. I wrote this email to seek your advice about taking the Analysis I (Real Analysis I), Abstract Algebra I, and Linear Algebra with Proofs. At...

Homework Statement Let F be a field. Show that F is isomorphic to F/{0} Homework EquationsThe Attempt at a Solution By the first ring isomorphic theorem, kernel of the homomorphism is an ideal which is either {0} or I. Hence F isomorphic to F/{0} I think I misunderstood the problem can...

Homework Statement Show that the group of units in Z_10 is a cyclic group of order 4 Homework EquationsThe Attempt at a Solution group of units in Z_10 = {1,3,7,9} 1 generates Z_4 3^0=1, 3^1=3, 3^2=9, 3^3= 7, 3^4= 1, this shows <3> isomorphic with Z_4 7^0=1 7^1= 7, 7^2= 9 7^3=3 7^4=1, this...

Hi, Are Calculus I, II, III courses a prerequisite requirement for studying Abstract Algebra? I have read that Proofs and a willingness to work hard is. I am studying Logic and Set Theory and want to study Abstract Algebra in the distant future. I am focused on Foundational and Pure...

Homework Statement Let G be any group and a in G, define f: Z → G by f(n) = a^n Apply any isomorphism theorem to show that range of f is isomorphic to a quotient group of Z Homework EquationsThe Attempt at a Solution The range of f is a^n , then quotient group of Z is Z/nZ Apply the first...

I am planning to study the following pure mathematics areas (on my own) and wanted to know if this is the best sequence: 1- Formal Logic 2 -Philosophical Logic 3- Sentential Logic 4- Predicate Logic 5- Symbolic Logic 6 -Set Theory 7 -Pure Mathematics (Intro, Pure Math I and II and Hardy) -...

Hello, A couple of years ago I studied abstract algebra from Dummit and Foote. However, I was never able to gain the intuition on the subject that I would like from that book. I want to study the subject again, and I want to use a different book this time around - one that covers a lot of...

Hi I recently read a book called "The fundamental theorem of algebra" by Fine and Rosenberger. It focused specifically on polynomials, and proved the theorem using several fields of mathematics; Two of the proofs were algebraic. Abstract algebra has been very difficult for me; Mostly because...

Homework Statement Let, M={ (a -b) (b a):a,b∈ℝ}, show (H,+) is isomorphic as a binary structure to (C,+) Homework Equations Isomorphism, Group Theory, Binary Operation The Attempt at a Solution Let a,b,c,d∈ℝ Define f : M→ℂ by f( (a -b) (b a) ) = a+bi 1-1: Suppose f( (a -b) (b a) )= f( (c...

I found out I can pick up a second major in math should I elect to take a two semester sequence in abstract algebra. My first major is in chemical engineering. Right now, I plan on taking a two semester sequence in either: 1) probability with measure theory, 2) abstract algebra (Dummit and...

Let N be a normal subgroup of a group G and let f:G→H be a homomorphism of groups such that the restriction of f to N is an isomorphism N≅H. Prove that G≅N×K, where K is the kernel of f. I'm having trouble defining a function to prove this. Could anyone give me a start on this?