Category Archives: Algebraic number theory

Algebraic number theory – index

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 3. Diophantine equations 4. Groups and rings 5. Fields … Continue reading

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Cyclotomic fields, part 2

In our previous article on cyclotomic fields we were talking about why the Galois group G of ℚ(μn)/ℚ is isomorphic to (ℤ/nℤ)×, where n∈ℤ and μn is the group of nth roots of unity, the roots of xn-1=0 in some … Continue reading

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Roots of unity and cyclotomic fields

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 … Continue reading

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Splitting of prime ideals in algebraic number fields

*.overline {text-decoration: overline;} Our series of articles on algebraic number theory is back again. Maybe this time it won’t be so sporadic. Stranger things have happened. The previous installment, of which this is a direct continuation, is here. All previous … Continue reading

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Splitting of prime ideals in quadratic extensions of ℚ, part 1

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 … Continue reading

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Factorization of prime ideals in extension fields

In this installment of our series on algebraic number theory, we’re going to do two things. First, we’ll look at how a prime ideal of one ring of algebraic integers factors into multiple prime ideals in the ring of integers … Continue reading

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More concepts from ring theory

We are at an important juncture in our discussion of algebraic number theory. From here on out, the path starts to go uphill more steeply, with quite a bit more abstraction and technical complexity. I hope you’ll follow along anyhow. … Continue reading

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