Abstract Algebra Problem Sets with Solutions

ORIGINAL POST

I work in a math-related field, and have a Master's degree in CS. So, after working for some time I noticed my math knowledge was not up to my own standards... So I have been working a pair of years to read 2 books of Apostol Calculus in my free time and do all exercises there, as well as went through point-set topology, or Chapters 1-4 in Rudin. I thought my mathematical maturity has increased many times from this.

However, I decided to also take an abstract algebra course from Math department from a local university. So I did. It went pretty ok... But... I CAN'T SEE SIMPLE SOLUTIONS or they come SO HARD. And last week was the last straw... i actually never aimed to be a math professor, I clearly lack in intellectual capacity to achieve that. But i thought that I would be at least able to easily solve simple math problems due to my better understanding of math from all the work with analysis...

And then i got this exercise in abstract algebra...

"Let R be a PID (principal ideal ring). Show that every (non-zero) prime ideal is maximal." That's it.

I was trying to solve it for 10 hours TEN HOURS, split into two 5-hour periods into 2 days. And I FAILED. In fact, i tried proving this by contradiction, assuming there exists a maximal ideal which includes the prime ideal. Then i tried constructing ideals of the form {ax + b} which would include the prime ideal properly. i tried to invoke theorems from the lectures, and I even tried proving that R/(p) is a field! But EVERYTHING failed!

Moreover, other students actually had no problem with this question... I knew there must be a simple solution, but my ego did not allow me to ask for hints... Not able to tolerate this anymore after 10 hours, i googled it. Opened the proof. This one: https://yutsumura.com/in-a-principal-ideal-domain-pid-a-prime-ideal-is-a-maximal-ideal/

Read the first line: it said, assume there exists an ideal properly including your prime ideal... I IMMEDIATELY KNEW that the approach should be the one by contradiction or related... i did not check the rest of the proof - i just solved it immediately.

The problem is really simple... but now i cannot forgive myself for not seeing this obvious approach/solution... Am i really that dumb, or something is wrong with my head... Why cannot i see simple solutions? it is actually not uncommon when I can sometimes solve more complicated proofs that this, which i did in Analysis i self-studied.

Do you have any tips maybe how to improve my brain? Did you ever have similar situations?

Is it possible that it is just my genetic ability that is not on par?

How do i know this (vs maybe not doing my study practice correctly)?

2 replies

throwaway4275571 · 4 years ago

If it's a hard problem and you keep seeing new path to continue exploring, time is well-spent.

But if it's an easy problem and you spent 10 hours not getting anything, I think you're not even in the right mindset. Like, your mind aren't generating new ideas in that 10 hours. Take a break, take a walk, do different problems, or just do something else. Without it, your mind was probably stuck in a rut and keep following the old paths over and over. Come back to it later, when you have a fresh mind.

3 upvotes on reddit

study_ai · OP · 4 years ago

hmmm, maybe it really makes sense to have a longer rest between problem solving to reset the brain. Though here tried to solve it in 2 days...so not sure, but definitely a valuable tip!

1 upvotes on reddit

See 2 replies

r/math • [2]

Summarize

Are there infinitely many $g$- or $m$-primes? (Any help with these questions is appreciated.)

Posted by musescore1983 · in r/math · 9 months ago

mathoverflow.net

7 upvotes on reddit

1 replies

Helpful

Not helpful

1 replies

Almondo21 · 9 months ago

writing this for the achievement

-3 upvotes on reddit

See 1 replies

r/mathematics • [3]

Summarize

An abstract algebra book with solved solutions.

Posted by Qazeevi · in r/mathematics · 3 years ago

Please recommend me an abstract algebra book which has questions with solutions because I'm facing difficulty in solving problems and proofs and exams are not too far.

20 upvotes on reddit

12 replies

Helpful

Not helpful

https://drive.google.com/file/d/1ImSF-vvXgpGnDx-XDsgoyYuqJYnhr7gU/view?usp=share_link

12 replies

BenSpaghetti · 3 years ago

Comtempory abstract algebra by Joseph Gallian has complete solutions.

10 upvotes on reddit

Qazeevi · OP · 3 years ago

Yes thanks just heard of it!

4 upvotes on reddit

OneMeterWonder · 3 years ago

I’ll second this one. My undergrad group theory used it.

2 upvotes on reddit

[deleted] · 3 years ago

Try Adventures in Group Theory—at least some of the solutions are in there, and I’m positive a solution manual exists online

5 upvotes on reddit

Qazeevi · OP · 3 years ago

Solution manual? Any website?

1 upvotes on reddit

[deleted] · 3 years ago

Google “[name of book] solution manual pdf”

0 upvotes on reddit

Machvel · 3 years ago

dummit and footes abstract algebra textbook has a lot of examples in each section (the most that i have seen in any upper division level textbook), and a good variety of problems from not so hard (to get used to working with the content) to hard.

3 upvotes on reddit

scoco · 3 years ago

Pinter's A Book of Abstract Algebra has solutions to many of its exercises, or at least solid hints. The book is well recommended but the exercises aren't its strongest point. The full text is not hard to find if you want to take a look.

2 upvotes on reddit

Qazeevi · OP · 3 years ago

Are you taking about solved examples?

1 upvotes on reddit

scoco · 3 years ago

There are solved examples, but I'm talking about the exercises at the end of each chapter.

1 upvotes on reddit

tenebris18 · 3 years ago

Schaum's outlines. Although not perfect or even close to being called an actual math text, they can help you speedrun the material once to just get an idea of things.

3 upvotes on reddit

AcademicOverAnalysis · 3 years ago

Schaums is a Series of textbooks that give a cliff notes version of pretty much every subject under the sun. The contents can be of varying quality from topic to topic. Think of this as a small step up from the “for dummies” series.

3 upvotes on reddit

See 12 replies

r/numbertheory • [4]

Summarize

Number Theory Paper Submission

Posted by Outside_Term1468 · in r/numbertheory · 2 months ago

I have been working on a number theory problem for a while now, and was hoping to submit it to arXiv, but I do not have access to the archive for number theory. I also haven't been able to get a hold any professors that I know because of the summer time. Would someone be willing to look over the paper? I have written it up in LaTex, and feel as though I am very close to the final proof of the problem.

edit: updated link

3 upvotes on reddit

9 replies

Helpful

Not helpful

9 replies

Enizor · 2 months ago

I don't really like the notation for the sum of divisors \sum_1^n a (a isn't the sum variable), I would recommend either \sum_{a \divides p} a or \sum_{k=1}^n a_k .

In the proof of Lemma 2.4: > Assume that there is a p ∈ P with only four divisors. Such a number could only be constructed as the multiple of two primes.

Such a number p can also be a prime cubed.

Theorem 2.3: you do prove how to find (z, c ∈ C) from an element of B. You do not prove that this solution is unique. (you claim that the function is bijective but never proved injectivity)

Theorem 3.1, case 2: > It is known from Theorem 2.3 that by adding z to any element in C the corresponding element in B can be reached.

It is only known from Theorem 2.3 "there exists a z", that one can use to reach B. Not that defining z as the sum of the series in C will always make you reach B.

3 upvotes on reddit

Outside_Term1468 · OP · 2 months ago

I would say the notation is clunky sure.

I agree with your comment on Lemma 2.4. I had that in my notes, but over looked it for some reason. Good catch. But there is no prime^3 = 1 + prime + prime^2 as this would imply prime^3-prime^2= 1+prime. Which is not true. So I think lemma 2.4 still holds.

I am not sure that I agree with your comment on Theorem 2.3. If we have said that there are two set B and C with the same cardinality there must be a bijective function. So shouldn't showing either the subjective or the injective piece means that you know the other is implied through the cardinality. We know that the cardinality for the two sets must be the same because each set is defined such that each element in B and C corresponds one to one to each element in the set of perfect numbers. Maybe I am wrong on this though?

In theorem 3.1 there are two parts case 1 is to show that when you assume some even perfect number we can get to a series in C that must be equal to z. In case 2 what we are showing is that when you assume that a series in C that is equal to z. I think what you are getting at is: "Take some \sum_{1}^{n-2}a=z in C and assume that \sum_{1}^{n-2}a=z." should be: Take some \sum_{1}^{n-2}a in C and assume that \sum_{1}^{n-2}a=z."

Even if I haven't resolved all of your qualms in this comment, and we assume that I find a way for all of my theorems and lemmas hold do you agree with my path on conjecture 3.1?

1 upvotes on reddit

Enizor · 2 months ago

> So I think lemma 2.4 still holds.

Yes, the proof just wasn't complete :) .

> We know that the cardinality for the two sets must be the same because each set is defined such that each element in B and C corresponds one to one to each element in the set of perfect numbers. Maybe I am wrong on this though?

It's easy to map from B to C because you just remove the last term of the sum. It is not easy to go from C to B: which element do you add to find an element of B ? Is this element unique? Until you prove this unicity, you only proved Cardinal(B) >= Cardinal(C).

For theorem 3.1, it wasn't clear to me that z was still defined as the largest proper divisor of p. The proof now seems alright.

For your final "conjecture" (where you provide a proof? name it a theorem then): I don't agree with (between 25. and 26.)

> q | z(2+sum) and q ∤ z ; It follows then that q | 2+sum

if a | b.c and a ∤ b, it does not follows that a | c. Counterexample: 20 | 8*10 ; 20 ∤ 8 and 20 ∤ 10 .

1 upvotes on reddit

AutoModerator · 2 months ago

Hi, /u/Outside_Term1468! This is an automated reminder:

Please don't delete your post. (Repeated post-deletion will result in a ban.)

We, the moderators of /r/NumberTheory, appreciate that your post contributes to the NumberTheory archive, which will help others build upon your work.

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

1 upvotes on reddit

hanleywashington · 2 months ago

Between (24) and (25) you may have an error.

You say q divides zd (where d is given by some finite sum). And you say that q does not divide z. Let's assume all that is correct. From this, you conclude that q divides d. There is a gap in the argument here.

For this to be true you would have to know that q and z are coprime.

2 upvotes on reddit

Outside_Term1468 · OP · 2 months ago

If you recall we have said that q=2*\beta*z this means q>z therefore when we get to:

q + z(2 + finite sum) = qz if we divide both sides by q we are left with

1 + {z(2+sum)/q}=z we know that because q is greater than z; q must divide (2+ finite sum).

Am I wrong?

edit:

I am wrong. See Enizor thread.

1 upvotes on reddit

Deweydc18 · 2 months ago

I do hate to be the bearer of bad news but you neither have nor are you close to a correct solution.

1 upvotes on reddit

swehner · 2 months ago

Should prefect be perfect?

4 upvotes on reddit

Outside_Term1468 · OP · 2 months ago

Yes.

1 upvotes on reddit

See 9 replies

r/mathematics • [5]

Summarize

Problems needed

Posted by math_lover0112 · in r/mathematics · 5 months ago

Hey everyone, I need a math problem (or a few) to go on a rabbit hole on. Any branch of math is good, I just can't find any problems that hook me currently. Thanks in advance!!

6 upvotes on reddit

9 replies

Helpful

Not helpful

9 replies

starkeffect · 5 months ago

Look up old Putnam exams.

3 upvotes on reddit

PersonalityIll9476 · 5 months ago

or Math Olympiads. You can burn a lot of spare time thinking about them.

1 upvotes on reddit

Astrodude80 · 5 months ago

If you know a little set theory, prove or disprove the following: for all cardinals p and q, if p^2=q^2, then p=q.

>!hehehehehehe!<

>!okay that might be a little mean so here’s the answer. If you assume the statement is true, then the Axiom of Choice is automatically true, so in models where AC fails, the statement is actually false. See Jech “The Axiom of Choice,” Ch 11!<

3 upvotes on reddit

Astrodude80 · 5 months ago

Read my question again: “for all cardinals p and q,…” -1 is not a cardinal.

1 upvotes on reddit

4S0K9 · 5 months ago

One problem would be finding a efficient algorithm to determine whether 2 graphs are isomorphic.

5 upvotes on reddit

math_lover0112 · OP · 5 months ago

I will look into that!

3 upvotes on reddit

DeGamiesaiKaiSy · 5 months ago

Uhhh

5 upvotes on reddit

MathTutorAndCook · 5 months ago

I got 99 problems but a cent makes 1

2 upvotes on reddit

aroaceslut900 · 5 months ago

I heard of this problem called the Riemann hypothesis. Not many people have tried to solve it so it's probably not too hard

3 upvotes on reddit

See 9 replies

r/APStudents • [6]

Summarize

AP Calc Problem Sets for practice (with solutions)

Posted by turksvids · in r/APStudents · 2 years ago

Hi!

I've been making problem sets every day this year for students who want to practice AP Calc stuff as they learn it. I think the questions are about the same the level of the AP exam. My plan is to continue through to the end of my Calc AB course (which should be around mid-March). I'll likely not continue with BC stuff at this time because...I'm getting a little burnt out on this project.

I've compiled what's posted so far this into this document. Each page contains:

the problems for the day,
a link to the YouTube video solutions.
a QR code to scan to get to the solutions

If you'd rather just check out some of the videos, here's a link to the playlist compiling all of them.

Feel free to share with your classmates/teachers/anyone you think could benefit or find these useful.

I hope that some of you find them helpful! I'd love to hear from you if you do!

35 upvotes on reddit

1 replies

Helpful

Not helpful

1 replies

mytdogman1 · 2 years ago

I love you, and you are now my favorite person

5 upvotes on reddit

See 1 replies

r/learnmath • [7]

Summarize

Problem sets and solutions equivalent to university assignments?

Posted by NoLimitWraith · in r/learnmath · 4 years ago

There are many university-level lectures available on YouTube, but what about problem sets and solutions equivalent to what would be assigned in a rigorous university setting? I know that MIT OpenCourseWare has some, but typically the solutions are not provided.

Is my best option to buy a math textbook? That's fine, but I probably don't need to be doing every problem in the textbook. How can I structure my learning such that I get the same knowledge as I would if I were to pay for a university course?

2 upvotes on reddit

4 replies

Helpful

Not helpful

4 replies

cirosantilli · 1 year ago

I dream of a world leading level university where all course materials will be open, including exam questions, and in which anyone can pay to take an exam and get a certificate for it, without taking any courses. We will get one day, I believe.

1 upvotes on reddit

jammasterpaz · 4 years ago

I think a lot of example class questions are online as well, on lecturer's personal sites, and course overview sites.

You're kidding yourself if you think a text book and youtube is any replacement for a university maths course, even if you don't skip book problems.

1 upvotes on reddit

NoLimitWraith · OP · 4 years ago

Provided that you have all video lectures, problem sets and exams with solutions, and are able to use Google and Khan Academy when you get stuck (and hire a tutor if that fails), I honestly do not see why self-learning cannot serve as a replacement for a university math course.

4 upvotes on reddit

jammasterpaz · 4 years ago

Well no, of course not. Not having had the benefit of it you wouldn't do, would you?

Prove me wrong by all means.

1 upvotes on reddit

See 4 replies

r/EngineeringStudents • [8]

Summarize

Problem Sets

Posted by kabayongnakahelmet · in r/EngineeringStudents · 3 years ago

Hi! I'm currently struggling with integrals and I want to practice it more so I can be better at it. Where can I find problem sets for integral calculus?

5 upvotes on reddit

5 replies

Helpful

Not helpful

5 replies

g1lgamesh1_ · 3 years ago

There is a pdf online for free, it has 800 integrals solved, google it.

1 upvotes on reddit

Whyudodisbro · 3 years ago

Integrals have a huge variety in complexity and methodology in solving them. Make sure you concentrate on the methods taught to you in the module to give you an idea of the difficulty you need to be focusing on.

3 upvotes on reddit

rinkelronkel · 3 years ago

Maybe you can try Khan's academy :)

2 upvotes on reddit

NotTiredJustSad · 3 years ago

Your textbook is a good place to start.

7 upvotes on reddit

Noah122345 · 3 years ago

Ask in r/math or google practice problems. Also, videos on YouTube for extra help are good in my opinion since many tutors do detailed explanations.

4 upvotes on reddit

See 5 replies

r/compsci • [9]

Summarize

Automata theory problem set with solutions?

Posted by Zerotil · in r/compsci · 6 years ago

Anyone know if there is a problem set out there with solutions to them for theory of automata class?

I want to get a good grasp of the material by practicing a variety of problems.

Thanks <3

61 upvotes on reddit

7 replies

Helpful

Not helpful

https://www.youtube.com/watch?v=eqCkkC9A0Q4&list=PLEbnTDJUr_IdM___FmDFBJBz0zCsOFxfK

7 replies

Zerotil · OP · 6 years ago

Currently browsing google to find problem sets.

With solutions:

No Solutions:

8 upvotes on reddit

CoderMonkey123 · 6 years ago

Check out this Youtube playlist by Ravindrababu Ravula, where he solves dozens of example problems on Automata theory. (These free Youtube videos are only a sample part of the paid course that he offers on his own website, but still there is a lot of material even in just these free Youtube videos.)

He has a bit of an accent though, but if you can understand Indian accents, then the videos are really good.

1 upvotes on reddit

ghs180 · 6 years ago

https://www.cs.cmu.edu/~15251/schedule

I took this class at CMU, it was a great course diving into automata theory. Check it out.

2 upvotes on reddit

7Buns · 6 years ago

I keep getting a document not found error :-(

2 upvotes on reddit

cds501 · 6 years ago

Try https://www.cs.cmu.edu/~15251/schedule.html

3 upvotes on reddit

[deleted] · 6 years ago

Try the sisper textbook. You can find it online for free. It has solutions for a bunch of problems at the end of each chapter.

15 upvotes on reddit

TotesMessenger · 6 years ago

I'm a bot, bleep, bloop. Someone has linked to this thread from another place on reddit:

[/r/u_cheenario] Automata theory problem set with solutions?

 ^(If you follow any of the above links, please respect the rules of reddit and don't vote in the other threads.) ^(Info ^/ ^Contact)

1 upvotes on reddit

See 7 replies

r/askmath • [10]

Summarize

Linear Algebra problems with infinite solution sets

Posted by jssa1995 · in r/askmath · 3 years ago

45 upvotes on reddit

8 replies

Helpful

Not helpful

8 replies

jssa1995 · OP · 3 years ago

Hi, I was recently given this as an assignment. I'm not entirely sure how to solve this, I have tried finding the echelon form of both systems but im not entirely sure what to do after that. Any help is greatly appreciated.

1 upvotes on reddit

OneMeterWonder · 3 years ago

Once you find the RREF, you can choose a few variables and solve for them in each of the equations. Suppose (9) ends up being a rank 4 system, meaning the RREF has four 1s on the diagonal. Since it has seven variables, you could use the equations in the RREF to solve for x₅, x₆, and x₇ all in terms of x₁, x₂, x₃, and x₄.

Once you do that, you can write down the solution set as

{(x₁, x₂, x₃, x₄, x₅, x₆, x₇): for all x₁, x₂, x₃, x₄ &in; &Ropf;}

where you replace the last three variables by their equations in the first four variables. This then tells you every possible solution to the system (9).

3 upvotes on reddit

HorribleUsername · 3 years ago

Careful with your HTML literals. You missed two closing ;s. Also, reddit's a pain about which literals it'll accept. &in; doesn't work, but &isin; does. Neither &Ropf; nor &reals; does, so you'll need a numeric code for that, or just copy-paste the character from whatever you're looking it up on.

2 upvotes on reddit

LittlePresident · 3 years ago

I'm not entirely sure about what is demanded here, but the system a) has 7 variables with only 4 functions. So you can't solve it. Thus it has an infinit numbers of solutions. For b) you have 5 variables and 5 functions. This should be solvable with only one solution.

-1 upvotes on reddit

RoiPhilippe · 3 years ago

For b, coefficients for x1 and x2 are always identical. So, you might solve for x1+x2, but not for them separately => infinit numbers of combinations for x1 and x2

1 upvotes on reddit

barcastaff · 3 years ago

What does it mean by infinite solution sets? If I assume that it means that solution would not be fixed, then you can compute the rref, recognise the free variables, and write the rest of the variables as linear combinations of free variables.

2 upvotes on reddit

chinkpro · 3 years ago

I guess you have to use Gaussian elimination technique

7 upvotes on reddit

AutoModerator · 3 years ago

Hi u/jssa1995,

This is an automated reminder from our moderators. Please read, and make sure your post complies with our rules. Thanks!

If your post contains a problem from school, please add a comment below explaining your attempt(s) to solve it. If some of your work is included in the image or gallery, you may make reference to it as needed. See the sidebar for advice on 'how to ask a good question'.

Rule breaking posts will be removed. Thank you.

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

No bio available

Karma

Posts

Comments

DeGamiesaiKaiSy

No bio available

2421

Karma

Posts

5256

Comments

mytdogman1

No bio available

785

Karma

Posts

446

Comments

Astrodude80

No bio available

5087

Karma

Posts

1163

Comments