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I haven't taken abstract algebra yet, but I am curious about connections between number theory and abstract algebra. Do the proofs of things like Fermat's little theorem, the law of quadratic reciprocity, etc. rely upon techniques found in abstract algebra?

Thanks.

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Much of the origins of abstract algebra has roots in Fermat's Last Theorem. In particular ideals, domains, and so on. That's about all I remember from the course in algebraic number theory. –  Asaf Karagila Jun 2 '13 at 1:05

4 Answers 4

Historically, there are many results in number theory that were proved without modern algebra, but are now viewed algebraically.

For example, the Chinese Remainder Theorem (CRT) states that any system of congruences $$\begin{array}{rcl}x&\equiv& a_1 \pmod{b_1}\\x&\equiv& a_2 \pmod{b_2}\\ &\vdots& \\ x&\equiv& a_n \pmod{b_n}\end{array}$$ for which $b_1,b_2,\ldots ,b_n$ are pairwise coprime has a solution $x$, and that all such solutions are congruent $\mod {\prod_{i=1}^n b_i}$. The CRT was first proved by Sun Tzu$^\star$ around the year $300$.

It turns out that the Chinese remainder theorem is equivalent to the group theoretic statement that, whenever $b_1,b_2,\ldots ,b_n$ are pairwise coprime and $b=b_1b_2\cdots b_n$, $$\mathbb{Z}_{b}\cong \mathbb{Z}_{b_1}\times \mathbb{Z}_{b_2}\times \cdots \times \mathbb{Z}_{b_n}.$$ In the completely unbiased opinion of this finite group theorist, the above is a much more intuitive form of the CRT. In fact, seeing the theorem in this form inspired a generalization: if $I_1,I_2,\ldots, I_n$ are pairwise coprime$^{\star\star}$ ideals of a commutative ring $R$, and $I$ is the product of these ideals, then $$R/I\cong R/{I_1}\times R/{I_2}\times \cdots \times R/{I_n}.$$ With this generalization we can use the CRT in other commutative rings, like the Gaussian integers $\mathbb{Z}[i]$, polynomial rings $R[x]$, formal power series rings $R[[x]]$, and so on. All of these rings are objects of interest in algebraic number theory, a discipline which (obviously) emphasizes the blurry line between what is abstract algebra and what is number theory.


$\hspace{2pt}^\star\hspace{2pt}$ To my great disappointment, this was not the Art of War guy.

$^{\star\star}$ Here "pairwise coprime" means that $I_j+I_k=(1)$ for each $j\neq k$.

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(+1) nice answer, (+1) Art of War –  vadim123 May 11 '13 at 17:27
    
@AlexanderGruber My comment was slightly off: we may call $\{I_1,I_2,\dots,I_n\}$ pairwise coprime if $I_j+\cap_{k\neq j} I_k=R$ for each $j=1,\dots,n$. Then the statement (CRT) is that for any $(a_1,a_2,\dots,a_n),a_i\in R$, there exists $a\in R$ such that $a\equiv a_k \mod I_k$ for each $k=1,\dots,n$. –  Pedro Tamaroff Jun 2 '13 at 3:05

See for instance Elementary Number Theory: An Algebraic Approach by Bolker.

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Major results like the ones you cite have many proofs, including ones using abstract algebra, but also ones using combinatorics. See George Andrew's number theory book for an excellent example of the latter.

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Oh OK. I'm more interested on the abstract algebra side of number theory. Are there any good books on this topic? –  user60887 May 11 '13 at 15:39
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@user60887 I enjoyed these lecture notes though they assume a good bit of algebraic background. –  Alexander Gruber May 11 '13 at 16:41
    
oh thanks these notes are very interesting. –  user60887 May 11 '13 at 19:23

While you do not need abstract algebra to formulate many interesting number-theoretic problems, you will find it very useful in solving those problems. Many theorems in elementary number theory are very simple in their formulations, yet their known proofs require advanced algebraic machinery, for instance Fermats Last Theorem. Personally I find algebraic proofs of many elementary number theory statements much more natural than their "elementary" proofs, see for example Fermats Small Theorem or Chinese Remainder Thorem and their proofs. Once you get some training in basic group theory and basic ring theory those proofs are very natural.

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