[Algebraic NT] Chapter 48

Question 1

How are two ideals added?

Answer 1

By appending the list of generators for both ideals

Question 2

How are two ideals multiplied?

Answer 2

By creating a new ideal with generators the pairwise products of the two original ideals (a1b1, a2b2, etc.)

Question 3

When is ideal multiplication equivalent to arithmetic multiplication?

Answer 3

When principal ideals are being multiplied

Question 4

When does an ideal divide another ideal?

Answer 4

When a is a superset of b, we say a divides b

Question 5

What is a prime ideal?

Answer 5

An ideal which does not divide any other ideal (same as chapter 5 definition)

Question 6

What is an integrally closed domain?

Answer 6

Domains which contain exactly all the elements which are the roots of monic polynomials in itself

Question 7

What is a Dedekind domain?

Answer 7

An integral domain which is integrally closed, Noetherian, and has all prime ideals maximal

Question 8

What is an example of a Dedekind domain?

Answer 8

The set of integers

Question 9

What is meant by the statement “prime ideals divide rational primes”?

Answer 9

A ring of integers with a nonzero prime ideal has said ideal maximal, as well as containing a rational prime

Question 10

When does “unique factorization”, similar to the fundamental theorem of arithmetic, hold true?

Answer 10

Within a Dedekind domain

Question 11

What is the rigorous definition for “unique factorization”?

Answer 11

Any nonzero proper ideal of a Dedekind domain can be uniquely written as a finite product of nonzero prime ideals

Question 12

What is meant by two ideals dividing each other, defined using unique factorization?

Answer 12

a | b implies that v_p(a) <= v_p(b), where v_p(n) denotes the amount of times prime ideal p appears in the unique factorization of n (similar to classical p-adics)

Question 13

What is the colloquial name of the Dedekind-Kummer theorem?

Answer 13

The ”Factoring Algorithm“

Question 14

What are the conditions of the Dedekind-Kummer theorem?

Answer 14

The theorem factors (p) for prime P in Z, where p does not divide the degree of the number field it is factored in.

Question 15

How is the Dedekind-Kummer theorem applied in the number field O_k : Z[a]?

Answer 15

1) reduce the minimal polynomial of a into (x-m)(x-n), …, modulo p
2) substitute a as x in the above polynomial
3) read the result as ideals generated by a-m, a-n, etc

Question 16

When does the Dedekind-Kummer theorem apply in monogenic fields?

Answer 16

As the degree is 1, for all primes

Question 17

In what type of structures do fractional ideals exist?

Answer 17

Dedekind domains

Question 18

What is a fractional ideal?

Answer 18

A set of the from 1/x * a, where a is an integral ideal (non-fractional), and x is in the Dedekind domain (note this includes integral ideals)

Question 19

What type of structure do the fractional ideals form?

Answer 19

A multiplicative group, with inverses a and a^(-1)

Question 20

What is another name for the ideal norm?

Answer 20

The absolute norm

Question 21

How is the ideal norm notated?

Answer 21

N(a) for a norm a

Question 22

How is the ideal norm defined?

Answer 22

|O_k/a|, where a is the ideal mentioned in N(a) and O_k is the number field in which a lies

Question 23

For what type of ideals is the ideal norm defined?

Answer 23

Nonzero ideals

Question 24

What type of arithmetic function is the ideal norm?

Answer 24

Completely multiplicative, can be extended to fractional ideals

Question 25

What is the ideal norm for a principal ideal?

Answer 25

The norm K/Q of ideal a