[Algebraic NT] Chapter 47 Flashcards
What is a Galois conjugate?
All the complex roots of the minimal polynomial for some algebraic number a
What is a norm, defined using linear algebraic concepts?
The determinant of the vector space map x to ax
What is a trace, defined using linear algebraic concepts?
The trace (sum of diagonal values) of the vector space map x to ax
What is a norm, defined using Galois conjugates?
The kth power of the product of Galois conjugates of a (where k = deg K/deg a)
What is a trace, defined using Galois conjugates?
k times the sum of the Galois conjugates of a (where k = deg K/deg a)
The norm is what type of arithmetic function?
Multiplicative
The trace is what type of arithmetic function?
Additive
What is a ring of integers?
The algebraic integers in K
What is the ring of integers of Q(sqrt(3))?
Z[sqrt(3)]
What is the ring of integers of Q(sqrt(d))?
Z[sqrt(d)] if d is congruent to 1 or 2 mod 4, Z[(1+sqrt(d))/2] if d is congruent to 3 mod 4
How is a ring of integers denoted?
O_k, where the O is in cursive script
What type of basis does a ring of integers have?
A Z-basis of algebraic integers
What is the relationship between algebraic numbers and algebraic integers?
An algebraic integer is always a whole number times an algebraic number
What is a monogenic number field?
Number fields with a basis consisting of powers of a single algebraic number, i.e. {1, a, a^2, …}
What is an example of a monogenic number field?
A quadratic field
