Generalization of algebraic numbers

If we restrict ourselves to [; \mathbb{Q} ;] algebraic numbers are [; a \in \mathbb{C} ;] such that there is a polynomial [; f \in \mathbb{Q}[X] ;] such that [; f(a)=0 ;] , this forms the field of so called algebraic numbers. But what if we generalize this idea to power series? Maybe something like: [; a \in \mathbb{C} ;] is called " [; \infty ;]-algebraic" if there exists [; f \in \mathbb{Q}[[X]] ;] such that [; f(a)=0 ;] converges to zero in the [; \mathbb{C} ;] with the usual metric. This kind of definition poses some convergence questions and loses a whole lot of nice properties but it's still interesting. For example [; \pi ;] is [; \infty ;] -algebraic since [;\sin(\pi)=0 ;] . Overall you get some interesting questions: like do the [; \infty ;] -algebraic numbers of [; \mathbb{C} ;] form a field? (they seem to form a commutative ring) Are there complex numbers which are not [; \infty ;] -algebraic ?

Is there any work done on this or is this just a useless definition?

Edit: Thanks for your answers! It seems that these weird numbers turn out to be all the complex numbers

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algebraic trick for radicals and imaginary numbers – 4 quick examples (relevant for ACT and SAT) youtu.be/USi-pLwXQ5I
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Algebraic Number Theory math buddy

Hello, I'm planning on watching the lectures and doing the assignments from Kedlaya's algebraic number theory course ( https://math.ucsd.edu/~kedlaya/math204a/ ) starting on friday. If anybody is interested we can work together and discuss problems or topics on discord since you can use latex on there or on jam board (although i'll use my mouse to to write so don't expect the best handwriting from me).

The course started a while back, so I'm a bit behind starting just now. From my understanding the course will follow Neukirch 's book so we can also read and discuss topics from there. There will be a second course next year that Kedlaya will probably also post online, if not we can keep following the book.

Note: I'm also busy with other school projects, classes and my undergrad thesis so my schedule will only allow me to go at a normal pace, ie watch 2 lectures/1 assignment a week

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Algebraic number theory. Very beautiful. Exciting. math.colorado.edu/~kstang…
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This is my algebraic number appreciation post

Hiya, I'm a math undergrad and I'm currently in love with algebra, and so I thought it would be neat to make a little post about how much I love algebraic numbers. I talked to one of my friends about how they are just roots of integer polynomials and I said for quadratic irrationals you pick 3 integers "and get a super cool new number for free!" Another really cool thing is that algebraic numbers are countable, and they form a field. Anyways, just showing the best numbers some love, ciao~

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[Integrated math 2:algebraic equations]I don’t even know where to start on number 1
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I’m trying to turn this complex number modulus equation into an algebraic expression which describes the set that it displays but I’m really stuck

|z-i| = 2|z+i|

This is what I have done

I let z = x+iy

|x+iy-i| = 2|z+iy+i|

Sqrt(x^2 + y^2 - 2y + 1) = 2 (sqrt(x^2 + y^2 + 2y + 1))

x^2 + y^2 - 2y + 1 = 4(x^2 + y^2 + 2y + 1)

3x^2 + 3y^2 + 10y= -3

Thanks!

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Looking to apply to Number Theory/Algebraic Geometry PhD programs, need help

Sorry if this is the wrong sub, if so, please direct me to the correct one!

So I’m looking to apply to grad school for NT/AG for Fall 2021 and I need help finding a cohesive list of universities I should be looking at. Obviously it’s easy to find top institutions on Google but I’m more interested in β€˜mid-tier’ universities that I have a higher shot of getting into. Thanks!

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Currently looking at Number Theory or Algebraic Geometry for graduate school? What undergrad topics should I be most familiar with for these two areas?
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Interesting animated visual for algebraic numbers and random polynomials community.wolfram.com/gro…
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Suggestion for Algebraic Number Theory

What books will you suggest for Algebraic Number Theory for an undergrad? Do I need any prerequisites other than Ring Theory and Field Theory? What are some of its applications?

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Interesting animated visual for algebraic numbers and random polynomials community.wolfram.com/gro…
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Needing help in proving that two numbers are algebraic, from given products of them

Hi all! I'm having trouble finding a solution for this:

I have two numbers. Let's call themΒ xΒ andΒ y.

I know for example thatΒ x^3 * y^7 Β is algebraic. We need to prove (or disprove) thatΒ x andΒ y are algebraic on their own. We knowΒ 4Β more, similar products of them. For example I have the same x and y but on different powers. And that's also algebraic.

Other example that was given: I know that x^3 * y^7 and x^6 * y^2 are algebraic

Any ideas on how to start?

Was thinking about finding minimal polinoms and somehow prove with them. Or should I think in indirect ways?

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Algebraic numbers

I was reading up on quintic polynomials a bit and I got a little confused by some terminology I read given what I know. I knew ahead of time that for some reason it was impossible to create a β€œquintic formula” the way we have quadratic, cubic, and quartic ones. As I’m reading I see that more specifically it means not all roots can be represented β€œalgebraically”, meaning to say that not all roots of quintic equations can be represented using addition, multiplication, subtraction, division, and nth roots/powers. The terminology is a little weird since I know a transcendental number, or a β€œnon-algebraic” number is one which cannot be a solution to a polynomial of integer coefficients.

So it seems like we have two contradicting ideas being described as β€œnon algebraic”: Either things that aren’t solutions to integer polynomials, or things that can be solutions, and by context are solutions, to integer polynomials that can’t be expressed using roots or easier operations. Any clarification would be appreciated. Thanks.

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Some thoughts and questions about arithmetic/algebraic operations and numbers

So I've been showerthinking about this a bit, and I'm by no means well-read on this so have probably got a lot wrong - I just want to get people's thoughts / opinions / explanations on my line of thinking regarding operations and sets:Generally, we can consider the following binary operations: addition, multiplication and powers. From this we can do algebra on the following fields: N, Z, Q, R (ignoring C for this as I don't know enough).

Now each set has these binary operations defined slightly differently. If f: S x S -> S is a binary operation for some set S, then we can see that addition under N and addition under Z are technically not the 'same' operation even thought they appear to roughly behave the same way. For example

Addition for N is f: N x N -> N. This is 'nice' and closed.

Addition for Z is f: Z x Z -> Z. This is also fine, but it's a different operation, since the sets are different and there are elements of Z (namely additive inverses) that 'behave' differently to N.

We sometimes say that 'N is not closed under subtraction' but in my mind this sentence is a red herring because there are no elements in the operation f: N x N -> N that even support this operation. In this way it makes no sense to talk about subtraction purely in terms of N. To try and get around that, you could define a binary operation f: N x Z -> Z, but then that is not the same thing. You cannot define the binary operation of subtraction on N if the b.o. must satisfy f: S x S -> S, right?

For addition and multiplication, this goes all the way up to R. Provided add and multiply are mappings f: S x S -> S where S is N, Z, Q, R, then actually there's technically no violation of closure. But then I noticed something different for powers, which I'll try to draw out:

Set Closed under binary ops: Not closed under binary ops:
N +, *, pow
Z +, * pow (e.g. 2^(-1)) not in Z
Q +, * pow (e.g. 2^(1/2)) not in Q
R +, *, pow

I guess what I'm trying to say is, under the hood, if we consider +, * and powers to be binary operations, there's actually no concept of 'closure violation' for + or *, because the sets involved in each definition are separate and this is what causes their behaviour and those the operations themselves to be different. Multiplication of negative numbers has a different geometric behaviour to multiplication of natural numbers, they are different 'things'.

But for powers, it seems that Z and Q fail because f

... keep reading on reddit ➑

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A question about algebraic numbers

I have two algebraic numbers r,s, where r = p(s) for some polynomial p. Are there any known equalities or inequalities involving the algebraic degrees of r and s, and the polynomial degree of p?

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I'm leveraging my self-quarantine time to transcribe my Algebraic Number Theory lecture notes

As the title says, I'm taking some time to LaTeX my lecture notes from undergrad.

Here's a link to the file on Dropbox. Currently only up to the first 16 pages are mathematical content - the rest is part of the template that I used, which I kept around in case I needed some tips or reminders. I'll probably upload updated PDFs as I work through the notes, perhaps once or twice a week. I also changed some names and the university name to try and preserve some anonymity.

Speaking of, I've been using this excellent LaTeX template, along with a couple other packages to suit my notational needs (I can list these if you all would like them).

I was recently accepted to a graduate program researching number theory, so I thought it was prudent to start with my algebraic number theory notes. I took the course in 2015 (the penultimate year of my undergrad), so this process has mostly been a review of the material for myself. For that reason, these notes are far more detailed than my actual lecture notes, as filling in the details of proofs and completing parts of the lecture that were left as exercises are some of the best ways to study pure math.

I plan to do the same for my Galois Theory and Measure Theory notes if I have time / am bored enough.

But I figured that at least one other person out there can probably make use of these!

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Completion of the algebraic numbers

Hello,

Recently I thought about the fact that the "jump" between Q and R is bigger than the "jump" between N and Z or Z and Q, and that, in a sense, it's not very logical. With N, Z and Q, we just talk about algebraic properties and suddenly, boum, continuity. Why not go through the algebraic numbers instead ?

Following this reasoning, I naturally wondered if we obtained complex numbers by completing the field of algebraic numbers. What bothers me is that this field cannot be ordered because of i and that we cannot properly define a distance because we do not have the real numbers. I'm not familiar with topology so I do not know if there's alternatives.

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In a mathematics department who specializes in algebraic topology, what is the "average" number of professors?

I am following this thread from a while back

https://www.reddit.com/r/math/comments/azliu1/strongest_math_departments_in_algebraic_topology/

that discusses the strongest math departments in algebraic topology that aren't necessarily "top-tier." I am an undergraduate student who will apply to algebraic topology programs next year. What is interesting to me is that schools like UT Austin, which is known for its topology department, seem to have at least five professors in algebraic topology and its related fields alone (such as K-theory, homotopy theory, etc). Then, we have schools like northwestern who have around two-three professors working in these fields. If you follow the thread above, there are more various departments they list with only one or two professors in the field. So I raise the question, how many professors working in algebraic topology and its related fields should exist on average to be considered a department who has a concrete algebraic topology group? Obviously algebra and analysis are far more popular in terms of modern mathematics and those fields have a tremendous amount of professors no matter where you go (I find departments with an analysis group of 10+ professors, but never an algebraic topology group of even 6+ professors). I mean, if I already have a professor I want to work with, then applying to a school with just that single professor shouldn't be an issue. But, I know I want to pursue this field and I want to have options when picking an adviser in the second or third year of my Ph.D. program. Of course, I should look into them before I even apply, but as I create relationships with them, I wouldn't be surprised if my initial favor to work with them is different later on. If I want to work in algebraic topology, should I be satisfied with a department with only two professors in this field? Or rather, is it common to go to a department with only two professors in algebraic topology? Thank you in advance.

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[Grade 10 Math: Algebraic factorising] I’m stuck with 9x^2+3x-4x^2, what two numbers equal 3 and when multiplied equal -36?
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Help with this question? When I plug in a number I get A, but the algebraic way to solve leads to choice D.
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"Algebraic number theory is a special case of group theory"

This sentence is on the wikipedia page for group theory, under applications of group theory.

Now, I feel like this sentence is not correct. Yes, group theory is used alot in ANT, but I do not see how it could be considered a special case in any sense. Do people agree?

Now, two subsections before, they had mentioned Galois theory. I think it would make a lot more sense to say that ANT is a special case of Galois theory. Maybe that is what they meant?

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πŸ“…︎ Sep 22 2019
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J. R. R. Tolkien's worldbuilding is like the algebraic number theory of fantasy literature. His story telling is like the pre-algebra.

Tolkien's worldbuilding is complex, deep, and incredibly cool. It's fascinating, and pretty damn genius. But if you don't know what you're getting into, it'll bend you over, smack you on the ass, and kick you out the door. His story telling on the other hand is much simpler. It's easy, it's fun, and while it hints at something much greater and more complex, at the end of the day, it's really meant for children. Don't get me wrong, there isn't anything wrong with that. I just feel like a lot of people equate the world he made to the stories he told.

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Is it possible to multiply 2 relatively prime algebraic irrationals and come up with a rational number? Are there any known examples?

I was just musing on how there is an infinite array of pairs of irrational numbers whose product is 2, but only one pair which I can actually name : sqrt2 and sqrt2. I'm curious if we know of any pair of algebraic irrationals, which aren't clear multiples of eachother, whose product is 2, or any rational.

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πŸ“…︎ Oct 23 2019
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What is the difference between analytic and algebraic number theory?

My university offers only one of them during two years (then it changes to the other) and I would like to know the differences of them and if its preferable to choose one or the other.

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πŸ“…︎ Jun 16 2019
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Algebraic notation(number and letters on the side of board) problem

Hello guys, I have a really big problem and I do not think I am the only one.

I have been playing online chess now for like 1 year, and I think i am somehow good at the moment(1600 elo bullet[i know it is not that much]), but this doesn't matter.

The problem is that I can't see the number and letter from a square, for example (A6, A2, etc.), i can't memorise it.

Is there anyway I could improve this side? I really want to learn the squares with their names and numbers but i just can't. I always have to look at the side of the board to see the letter and number coordinated to the square.

Any tip is very much apreciated, thank you guys!

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Can all real algebraic numbers be represented as an expression of roots of whole numbers(nestled if necessary)?

And if so, is it legitimate to view this as the primary thing that distinguishes between algebraic and transcendental numbers: That given any set of algebraic numbers we can always correctly order them in magnitude for certain, but given any list of transcendental numbers we may not always be able to order them in magnitude for certain?

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πŸ“…︎ Apr 12 2019
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Computational algebraic number theory

Hello r/math !

I'm a college student and my semester on ``basic" algebraic number theory course has started. The most `exciting" part of elementary number theory to me is that you can churn out some nice number theoretic calculation in c++/python and have some nice conjectures and then you can prove it theoretically which is really nice ! Or you can just fiddle around with them to have some intuition which is really valuable in general.

So I wanna do the same for algebraic number theory. Can you people recommend some software/books so that I calculate basic ANT stuff like for example checking whether the ring of integers of a number field is UFD or not, if it's UFD then checking how primes in O_Q factor over O_K, or for example when L is an extension of K then how primes in O_K factorizes in O_L, ``density" of primes in that ring of integers etc etc ?

I tried Mathematica but then some people suggested that it won't be fit for my use, and then some other people suggested SageMath but I don't know how to use it for such purposes (the documentation looks to boring for me to read fully). I also know C++/Python but I don't know how to implement the algorithms for doing so (and the algorithms themselves are probably too tricky/time consuming to figure out by myself; and I am not sure whether figuring out the algorithms by myself would be a productive effort or not).

All in all: I want to know some programming languages/algorithms which are easily implemented by some non-CS student to be able to perform computational ANT experiments to have some nice conjectures and some concrete intuition in ANT.

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πŸ“…︎ Sep 19 2019
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[video] The Future of Mathematics? (a great talk by Kevin Buzzard, an algebraic number theorist, about automated theorem proving in Lean) youtube.com/watch?v=Dp-mQ…
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πŸ“…︎ Dec 26 2019
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Doubly Algebraic Numbers?

If algebraic numbers are defined as numbers that are a solution to some polynomial with integer coefficients, what about polynomials with algebraic coefficients? Does this set have a name?

What if you continue this procedure, making a set out of polynomials with coefficients from the previous sets, would this eventually contain all the computable reals?

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πŸ“…︎ Aug 08 2018
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Preparation for Algebraic number theory

What should I know, and especially what should i be fluent in before taking algebraic number theory ?

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πŸ“…︎ Mar 29 2019
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