Verified answer

An algebraic integer is a complex number that is a root of a polynomial with leading coefficient 1, (i.e. monic). Examples are 5 + 3√2, 1 − 3i, and (1 + i√3)/2

The sum, difference and product of algebraic integers are again

algebraic integers - that is what is meant by a closed field.

An algebraic integer of the form a + bsqrt(D) where D

is "squarefree", forms a quadratic field and is denoted Qsqrt(D).

It is real if D > 0 or imaginary if D < 0

Keep a copy of the following arithmetic rules

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Arithmetic in quadratic number fields

Addition and subtraction

[a + bsqrt(D)] +/- [c + dsqrt(D)] = (a +/- c) + (b +/- d)sqrt(D)

Multiplication,

[a + bsqrt(D)]*[c + dsqrt(D)] = (ac + bdD) + (ad + bc) sqrt(D)

Division

[a + bsqrt(D)]/[c + dsqrt(D)] = [(ac - bdD) + (bc - ad)sqrt(D)]/(c^2 - d^2D)

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Unique factorizations of algebraic integers in quadratic fields

Not all algebraic integers have a unique factorisation. For example as Dedekind pointed out

6 = [1 + sqrt(-5)][1 - sqrt(-5)] or the more obvious 2*3. He wanted to devise a smaller sub-group of algebraic integers, (a domain or ring), which would have the unique factorisation we are used to with ordinary integers

The ideas of the theory of ideals and several lemmas in number theory needed mean that there is no "one line answer" to "how do you determine if algebraic integers have unique factorization.

You may wish to read this entertaining ramble on algebraic integers

http://uqu.edu.sa/files2/tiny_mce/plugins/filemana...

or this more formal paper

http://www.ams.org/journals/bull/1924-30-07/S0002-...

If you find a short and simple guide to unique factorization of algebraic integers please post it here.

I hope this at least gets you started.

Regards - Ian

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