Algebraic integers are complex numbers that are roots of monic polynomials with integer coefficients. They form a subring of complex numbers and play a crucial role in number fields, extending beyond rational integers to include non-rational algebraic integers.
Minimal polynomials are the monic polynomials of least degree with integer coefficients that have an algebraic integer as a root. They're always irreducible over rational numbers and their degree equals the field extension degree generated by the algebraic integer.
Algebraic Integers and Properties
Definition and Fundamental Characteristics
- Algebraic integers comprise complex numbers serving as roots of monic polynomials with integer coefficients
- Form a subring of complex numbers, exhibiting closure under addition and multiplication
- Encompass all rational integers while extending beyond to include non-rational algebraic integers
- Preserve algebraic integer status under addition and multiplication operations
- Constitute a finitely generated Z-module within the ring structure of a number field
- Possess rational integer values for their trace and norm
- Exhibit divisibility properties analogous to rational integers (divisibility rules, prime factorization)
Mathematical Structure and Operations
- Generate a ring structure within a number field, allowing for algebraic operations
- Closure property ensures sum and product of two algebraic integers yield another algebraic integer
- Trace of an algebraic integer $α$ calculated as sum of its conjugates: $Tr(α) = α_1 + α_2 + ... + α_n$
- Norm of an algebraic integer $α$ computed as product of its conjugates: $N(α) = α_1 * α_2 * ... * α_n$
- Satisfy multiplicative property of norms: $N(αβ) = N(α)N(β)$ for algebraic integers $α$ and $β$
- Form ideals within their ring, enabling the study of ideal theory in algebraic number fields
- Allow for generalization of unique factorization through ideal decomposition in Dedekind domains
Minimal Polynomial of an Algebraic Integer
Definition and Properties
- Represents the monic polynomial of least degree with rational integer coefficients having the algebraic integer as a root
- Always irreducible over rational numbers, ensuring no factorization into lower degree polynomials with rational coefficients
- Degree equals the degree of field extension generated by the algebraic integer over rational numbers
- All roots constitute conjugates of the given algebraic integer, forming a complete set of algebraically related elements
- Coefficients expressible through elementary symmetric functions of the conjugates of the algebraic integer
- Discriminant of minimal polynomial provides crucial information about the field extension (ramification, splitting behavior)
- Serves as the defining polynomial for the number field generated by the algebraic integer
Computation and Applications
- Found using characteristic polynomial and rational canonical form of matrix representation for multiplication by the algebraic integer
- Characteristic polynomial $p(x)$ of a matrix $A$ defined as $p(x) = det(xI - A)$, where $I$ denotes the identity matrix
- Rational canonical form simplifies the matrix to block diagonal structure, revealing the minimal polynomial
- Utilized in determining the Galois group of the field extension generated by the algebraic integer
- Plays crucial role in analyzing the splitting field and algebraic closure of number fields
- Aids in computing integral basis and discriminant of number fields
- Facilitates the study of ramification and decomposition of primes in algebraic number fields
Ring of Integers in a Number Field
Fundamental Concepts and Characterization
- Number field defined as finite extension of rational numbers, denoted $K = \mathbb{Q}(α)$ for some algebraic number $α$
- Ring of integers $O_K$ consists of all elements integral over the integers within the number field
- Every element of $O_K$ qualifies as an algebraic integer, satisfying a monic polynomial with integer coefficients
- Conversely, all algebraic integers within the number field belong to $O_K$, ensuring completeness
- $O_K$ forms an integrally closed domain within the number field, crucial for ideal theory
- Represents a Dedekind domain, enabling unique factorization of ideals into prime ideals
- Integral closure property of $O_K$ in $K$ forms the basis for studying arithmetic in algebraic number fields
Proof Techniques and Implications
- Proof involves demonstrating $O_K$ as integrally closed in the number field $K$
- Utilizes the fact that $O_K$ serves as integral closure of rational integers in $K$
- Employs techniques from commutative algebra, including properties of integrally closed domains
- Highlights connection between algebraic integers and integral elements in ring extensions
- Establishes foundation for studying ideal class groups and unit groups in number fields
- Facilitates investigation of prime ideal decomposition and ramification in algebraic number theory
- Crucial for understanding more advanced topics (class field theory, Diophantine equations)
Algebraic Integers vs Integral Basis
Integral Basis: Definition and Properties
- Represents a basis for the ring of integers $O_K$ of a number field $K$ as a Z-module
- Consists of algebraic integers generating all other algebraic integers in $K$ through integer linear combinations
- Guarantees existence due to finiteness of $O_K$ as a Z-module (finite rank free abelian group)
- Discriminant of integral basis relates to discriminant of the number field, providing important invariant
- Number of elements in integral basis equals degree of number field extension over rationals
- Allows representation of any algebraic integer $α$ in $O_K$ as $α = a_1ω_1 + a_2ω_2 + ... + a_nω_n$, where $ω_i$ form integral basis and $a_i$ integers
- Plays crucial role in computations involving ideals and factorization in algebraic number fields
Relationship and Computational Aspects
- Finding integral basis equivalent to determining complete ring of integers in number field
- Integral basis elements themselves algebraic integers, but not all algebraic integers part of basis
- Techniques for finding integral basis include analyzing trace form and index of submodules in $O_K$
- Trace form defined as bilinear form $T(x,y) = Tr(xy)$ for $x,y$ in $O_K$, used to construct integral basis
- Index of submodule $[O_K : Z[α]]$ helps identify additional algebraic integers to include in basis
- Computation often involves successive approximation, refining potential basis elements
- Integral basis crucial for practical computations in algebraic number theory (ideal factorization, class group computation)