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)