Monthly Archives: March 2013

6 – Algebraic integers, unique factorisation, and Fermat’s last theorem

(This is part of my series on algebraic number theory.)

The foundations of what is now algebraic number theory were devised in the late 19th century in various attempts to prove Fermat’s last theorem, which states that the equation
\[ x^n+y^n = z^n \]
has no solution in positive integers when $n > 2$. This post will give the first part of an attempted proof by Lamé in 1847, using unique factorisation in certain subrings of number fields.

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5 – Computing in number fields

(This is something of a computational aside in my series on algebraic number theory.)

Recall that a number field is a field of the form $\mathbf{Q}(\omega)$, where $\omega$ is a complex number which is algebraic over $\mathbf{Q}$ (that is, is a root of some polynomial in rational coefficients). Then the elements of $\mathbf{Q}(\omega)$ are of the form
\[ a_0 + a_1\omega + a_2\omega^2 + \dots + a_{n-1}\omega^{n-1}, \]
where $n$ is the degree of $\omega$ over $\mathbf{Q}$ (that is, the degree of a monic, irreducible polynomial in rational coefficients which has $\omega$ as a root).

What this means is that an element of a number field can essentially be viewed as a polynomial in rational coefficients, and in turn this means that we can easily perform the operations of addition and multiplication in any number field (indeed, in any field extension, provided we can easily compute in the base field). This post will demonstrate how this is done, both by hand when the number field is manageable, and with a computer using PARI/GP. The standard reference for computational matters in number fields (and many others) is the book by Cohen.

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3 – Unique factorisation

(This is part of my series on algebraic number theory.)

Since number theory is ultimately concerned with problems in the ring of integers $\mathbf{Z}$, we try to find rings which share as many properties with $\mathbf{Z}$ as possible. In the previous post we have seen commutative rings with unity, which share two key properties of $\mathbf{Z}$. In this post we present two additional properties of $\mathbf{Z}$ which are shared by the rings we will be ultimately interested in. Of course, the title of this post gives a hint of what we are after. ;) In this post and all the subsequent ones, when we speak of a ring, we mean a commutative ring with unity.

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2 – Rings

(This is part of my series on algebraic number theory.)

We have seen that most of the number systems we are familiar with (the rational, real, and complex numbers) are fields. There is one notable exception: the set $\mathbf{Z}$ of integers. Indeed, if you try to tick all the boxes in the definition of a field, you see that $\mathbf{Z}$ does not fulfill all the requirements. It comes close, however: the only requirement it fails to fulfill is the existence of multiplicative inverses for all its elements. Perhaps then it might be worthwhile to relax our requirements a little and consider sets which have the same properties as $\mathbf{Z}$. This post deals with such sets, called rings.

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Algebraic number theory

This is a series of posts on algebraic number theory which builds upon my previous series on field theory. The goals are mostly the same: to give the interested reader with only a modest background in mathematics a first “feel” of the subject, and perhaps to motivate further study. I have tried to also keep the same style of exposition, but the pace will be somewhat faster, and might be difficult to follow for the reader who has merely read my previous series and not otherwise done a lot of work with the basic objects of algebra (groups, rings, fields, polynomials).

For the (relative) beginner wishing to study algebraic number theory I warmly recommend Marcus’s book (and indeed much of what I will write in this series will be based on it). At the beginning, it requires only a very modest background in abstract algebra (having studied everything in Fraleigh’s book will more than suffice), and the level rises progressively, which makes it possible to study it in parallel with a more advanced course in abstract algebra.

The posts are as follows:

1 – R vs. Q and number fields
2 – Rings
3 – Unique factorisation
4 – Cyclotomic fields

Linux smart card authentication – PAM

(This is part of my howto on smart card authentication in Linux.)

PAM (Pluggable Authentication Modules) is an authentication framework which uses modules to authenticate users using a wide variety of methods. A PKCS#11 PAM module exists, which allows us to use smart cards to authenticate against any service which uses PAM. The most obvious usage of PAM is system logins, either console or graphical, but a lof of other services, for example sudo, use it (you can have a look in /etc/pam.d to see all currently installed services which use PAM).

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Linux smart card authentication – OpenSSL

(This is part of my howto on smart card authentication in Linux.)

You can use the private key stored on your card with OpenSSL, just like you would use an on-disk key. Among other things, you can sign files, decrypt files encrypted with your public key, or generate X.509 certificates for your key. Since this is not an OpenSSL guide, I will not describe those operations in detail, you can refer to the OpenSSL page in the Ubuntu Server Guide if you are not familiar with them, the syntax is the same (except for the necessary command flags to tell OpenSSL to use your smart card, see below).

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