Tags » Algebraic Integer
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
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