Lecture notes regarding integers ?

[garyljc]

Hey guys ,
was wondering if you guys know of any lecture notes regarding integers ?
i would like to further my knowledge in this field ... cheers

 

[HallsofIvy]

Are you serious? There are thousands of different "fields" involving integers. You will have to narrow down the search! What, exactly, are you looking for? Number theory problems such as solve Diophantine equations? Analysis questions such the the definition of integers and basic properties?

 

[CRGreathouse]

While you're at it, HallsofIvy, how about some notes on real numbers too?

 

[garyljc]

I guess to start with I would like to look at the basics such as the proof of : If S ⊂ ℤ , then there is a natural number g such that S={ gn : n ∈ ℤ }

And also i came across a proposition somewhere in some maths books ( but it's only a small part ) Saying that :
- If A and B are subgroups of ℤ , then so is their intersection A ∩ B and so is the set
{m+n : m ∈ A , n ∈ B }
they followed on by saying this which i don't really get the picture
- A ∩ B contains every subgroup contained in both A and B
- A + B is contained in every subgroup containing both A and B

someone please help me out =) thanks

 

[HallsofIvy]

I guess to start with I would like to look at the basics such as the proof of : If S ⊂ ℤ , then there is a natural number g such that S={ gn : n ∈ ℤ }

And also i came across a proposition somewhere in some maths books ( but it's only a small part ) Saying that :
- If A and B are subgroups of ℤ , then so is their intersection A ∩ B and so is the set
{m+n : m ∈ A , n ∈ B }
they followed on by saying this which i don't really get the picture
- A ∩ B contains every subgroup contained in both A and B
- A + B is contained in every subgroup containing both A and B

someone please help me out =) thanks

You are using non-standard fonts that do not display on my web-reader. Please try LaTex or just stating the problem in words.

 

[CompuChip]

I guess to start with I would like to look at the basics such as the proof of : If $S \subset \mathbf{Z}$ , then there is a natural number g such that $S = \{ gn : n \in \mathbf{Z} \}$.

Am I misreading this, or is this simply not true? For example, take S = {1, 2, 3}, this is finite, while any set of the form $\{ g n : n \in \mathbf{Z} \}$ is necessarily {0} or countable infinite.

And also i came across a proposition somewhere in some maths books ( but it's only a small part ) Saying that :
- If A and B are subgroups of $\mathbf{Z}$ , then so is their intersection A ∩ B and so is the set
$\{m+n : m \in A , n \in B \}$
they followed on by saying this which i don't really get the picture
- $A \cap B$ contains every subgroup contained in both A and B
- A + B is contained in every subgroup containing both A and B

You can also just check the group axioms and see that it is true.

 

[garyljc]

this is what i copied exactly from the book ...
but anyways , do you know of any sites that has such notes or reading material that could help me ?

 

[garyljc]

by the way i did make a mistake
If S ⊂ ℤ is a subgroup , then there is a natural number g such that S={ gn : n ∈ ℤ }
this should be the right one

halls , where could i get latex ?

 

[HallsofIvy]

You don't have to "get it" at all, it's part of this forum. Start with [ tex ] (without the spaces) and end with [ /tex ] and use LaTex syntax in between. Here's an example:
$$e^x= \sum_{n=0}^\infty \frac{x^n}{n!}$$
Click on that to see the code.

More on LaTex syntax can be found here:
https://www.physicsforums.com/showthread.php?t=8997

 

[neutrino]

but anyways , do you know of any sites that has such notes or reading material that could help me ?

Check http://users.ictp.it/~stefanov/mylist.html" .

 

[garyljc]

thanks thanks ...
halls ... i'll get it done right away =)

 

[HallsofIvy]

I guess to start with I would like to look at the basics such as the proof of : If , then there is a natural number g such that .