[Algebraic NT] Chapter 46 Flashcards
What is the leading coefficient of a minimal polynomial?
one, as it is a monic polynomial
What set must coefficients of a minimal polynomials belong to?
The rational numbers
What is the minimal polynomial of x?
A monic polynomial with rational coefficients with lowest possible degree which has f(x) = 0
What is an algebraic number?
A number which has a minimal polynomial
What is an algebraic integer?
A number which has all coefficients of its minimal polynomial integral
What does Gauss’ lemma state?
The root of any polynomial with integer coefficients are algebraic integers
What is a number field?
A field Q(a1, a2, …, an) with elements in the form q + q1a1 + q2a2 + …
What type of structure is a number field, using linear algebraic concepts?
A Q-vector space
What does Artin’s theorem state?
Any number field is isomorphic to Q(k) for a single algebraic number k
What common set is equivalent to the set of rational algebraic integers?
The integers
What type of structure do the algebraic integers form?
A ring
What type of structure do the algebraic numbers form?
A field
