Analytic number theory is only one of many different approaches to number theory. Another important branch of the subject is algebraic number theory, which studies algebraic structures (e.g. 5,688 more words

## Tags » Algebraic Integer

#### Neukirch 1.2.1

Is an algebraic integer? (For simpler notation, we call this number .)

Short answer: Yes.

Long answer: We multiply by the denominator’s conjugate and simplify to get that… 54 more words

#### If A is an ideal of integers in an algebraic number field and if bA is contained in A, then b is an integer

Let be an algebraic number field with ring of integers , and let be an ideal. Suppose such that . Prove that .

Let be a basis for over . 61 more words

#### Show that a given algebraic integer is a unit in the ring of integers in the field extension it generates

Let be a root of . Show that, as ideals in the ring of integers in , .

Note that is a product of the minimal polynomials of its roots. 31 more words

#### In a quadratic field, the quotient of two algebraic integers with the same norm is a quotient of an algebraic integer by its conjugate

Let be a quadratic field and let be algebraic integers such that . Prove that there exists an algebraic integer such that . Give a nontrivial example. 57 more words

#### In a quadratic field, every element of norm 1 is a quotient of an algebraic integer by its conjugate

Let be a quadratic field, and let be an element with . Prove that there exists an algebraic integer such that , where denotes conjugate. 56 more words

#### A bound on the number of irreducible factors of an element in an algebraic integer ring

Let be an algebraic integer in an algebraic number field . Let denote the norm over . Prove that the number of factors in any irreducible factorization of over is at most . 51 more words