openquestions.com

mathophilia.wordpress.com

Groups and rings | Mathophilia

https://mathophilia.wordpress.com/2015/05/05/groups-and-rings

Fields and Galois theory →. May 5, 2015. In our previous installment of the series on algebraic number theory, we took a little detour into Diophantine equations. In order to provide some motivation for the theory itself. Prior to that, we had looked at different types of numbers. To give a perspective on the sorts of objects the theory deals with. In those discussions, we touched (explicitly or implicitly) on abstract algebraic structures called groups. G is a mathematical system consisting of a set of ...

mathophilia.wordpress.com

Mathophilia | Math is easy | Page 2

https://mathophilia.wordpress.com/page/2

Newer posts →. Splitting of prime ideals in quadratic extensions of ℚ, part 1. May 5, 2015. Our discussion of algebraic number theory returns by popular demand. Way back last April we presented some generalities on factorization of prime ideals in extension fields. (For explanation of what that means, including other necessary concepts, you’ll have to review earlier installments of this series, which can be found here. The discussion in the previous installment. So (p) splits completely. (e=f=1, g=2).

mathophilia.wordpress.com

Cyclotomic fields, part 2 | Mathophilia

https://mathophilia.wordpress.com/2015/05/05/cyclotomic-fields-part-2

Roots of unity and cyclotomic fields. Algebraic number theory – index →. Cyclotomic fields, part 2. May 5, 2015. In our previous article. On cyclotomic fields we were talking about why the Galois group G of ℚ(μ. 8474; is isomorphic to (ℤ/nℤ). Where n∈ℤ and μ. Is the group of n. Roots of unity, the roots of x. 1=0 in some extension of ℚ. (Check here. For a list of previous articles on algebraic number theory.). So far we’ve shown that G is isomorphic to a subgroup of (ℤ/nℤ). The group μ. Is a primitive n.

mathophilia.wordpress.com

Roots of unity and cyclotomic fields | Mathophilia

https://mathophilia.wordpress.com/2015/05/05/roots-of-unity-and-cyclotomic-fields

Splitting of prime ideals in algebraic number fields. Cyclotomic fields, part 2 →. Roots of unity and cyclotomic fields. May 5, 2015. In preparation for many good things that are to come, we need to have a talk about another important class of field extensions of ℚ – the cyclotomic extensions. (Check here. For a list of previous articles on algebraic number theory.). Since f(1)=0, x-1 is one factor of f(x), and f(x)/(x-1) = x. Roots of unity – denoted by ζ. For some integer k, 1≤k n. So the smalle...

mathophilia.wordpress.com

A brief history of algebra | Mathophilia

https://mathophilia.wordpress.com/2015/05/05/a-brief-history-of-algebra

The Riemann hypothesis: Dedekind zeta functions. Numbers – rational and irrational, real and imaginary →. A brief history of algebra. May 5, 2015. The place to begin, naturally, is with a brief history of algebra itself. The word “algebra” comes from Arabic: al-jebr. Muhammad ben Musa al-Khwarizmi seems to have been the first person whose writing uses the term al-jebr. To lead – hence to re- duce. For performing this simplification. Such methods are known as algorithms. And hence c = √(a. In addition to ...

mathophilia.wordpress.com

Modular arithmetic | Mathophilia

https://mathophilia.wordpress.com/2015/05/05/modular-arithmetic

Fields and Galois theory. Rings: fields of fractions →. May 5, 2015. After some hiatus, let’s return to our discussion of algebraic number theory. Check here. We have already talked briefly about groups and rings ( here. But because ring theory is so important to the whole subject, we need to go into it a lot deeper. As preparation for that, we’ll look at some of the simplest examples, which will turn out to be greatly generalized in the sequel. In any case, the construction is so important and pervasive...

mathophilia.wordpress.com

Factorization of prime ideals in extension fields | Mathophilia

https://mathophilia.wordpress.com/2015/05/05/factorization-of-prime-ideals-in-extension-fields

More concepts from ring theory. Splitting of prime ideals in quadratic extensions of ℚ, part 1 →. Factorization of prime ideals in extension fields. May 5, 2015. In order to understand what’s discussed here, you’ll need to recall a great deal of what’s been discussed before. You can find a list of all previous installments here. If you need to review. Suppose E⊇F is a field extension and O. Are the corresponding rings of integers. We have O. And if I is any ideal of O. Then of course I ⊆ O. An ideal of O.

mathophilia.wordpress.com

Rings of algebraic integers | Mathophilia

https://mathophilia.wordpress.com/2015/05/05/rings-of-algebraic-integers

Rings: fields of fractions. Rings and ideals →. Rings of algebraic integers. May 5, 2015. 8220;The time has come,” the Walrus said,. 8220;To talk of many things:. Of shoes–and ships–and sealing-wax–. Of cabbages–and rings. Well, that’s not exactly what he said, but close enough. In our last installment. Where we wanted solutions. That are integers, or “nearly” so. Abstractly, we know by the fundamental theorem of algebra that if n is the degree of f(x), then exactly (counting multiplicity) n solutions ex...

mathophilia.wordpress.com

Uniqueness of factorization | Mathophilia

https://mathophilia.wordpress.com/2015/05/05/uniqueness-of-factorization

Failure of unique factorization →. May 5, 2015. Time for another installment of the series on algebraic number theory. Check here. In this installment we’re going to look at an important property that some rings (such as ℤ) have, although most rings do not. But it is a useful and important property for proving many number theoretic results, which is why one bothers to consider it. We’ll illustrate that soon. But first we need a little terminology. In any ring, a unit. As a matter of notation, if one numb...

mathophilia.wordpress.com

Algebraic number theory – index | Mathophilia

https://mathophilia.wordpress.com/2015/05/05/algebraic-number-theory-index

Cyclotomic fields, part 2. Review of Mathematics without Apologies. Algebraic number theory – index. May 5, 2015. This is a list of posts in the algebraic number theory series to date, oldest first. 1 A brief history of algebra. 2 Numbers – rational and irrational, real and imaginary. 4 Groups and rings. 5 Fields and Galois theory. 7 Rings of algebraic integers. 8 Rings and ideals. 9 Uniqueness of factorization. 10 Failure of unique factorization. 11 Why do we care about unique factorization? Review of M...