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Semester 2: Advanced Algebra
Extension fields - Transcendence of e
Extension fields - Transcendence of e
Introduction to Extension Fields
Extension fields are essentially larger fields containing a given field. They provide a framework for studying polynomial equations and algebraic structures. By adding elements to an existing field, we can create new fields that incorporate the properties and structure of the original.
Definition of Transcendental Numbers
Transcendental numbers are numbers that are not roots of any non-zero polynomial equation with rational coefficients. The number e, which is the base of natural logarithms, is a well-known example of a transcendental number.
The Algebraic Closure of a Field
An algebraic closure of a field is an extension field in which every non-constant polynomial has a root. While the algebraic numbers form an algebraic closure, transcendental numbers like e exist outside this closure.
Proof of Transcendence of e
One of the most famous proofs of the transcendence of e was provided by Charles Hermite in 1873. The proof involves demonstrating by contradiction that e cannot be the root of any polynomial with rational coefficients, using properties of the exponential function.
Applications of Transcendental Numbers
Transcendental numbers have significant implications in various fields, including number theory, calculus, and cryptography. Understanding these numbers can provide insights into the nature of solutions to polynomial equations.
Examples of Extension Fields
Common examples of extension fields include the field of complex numbers as an extension of the real numbers and the field of rational numbers adjoined with the square root of a number. These extensions play key roles in algebraic and transcendental number theory.
Roots of Polynomials - More about roots
Roots of Polynomials
Definition of Roots
Roots of a polynomial are the values of the variable that make the polynomial equal to zero. For a polynomial P(x), if P(a) = 0, then a is a root.
Types of Roots
Roots can be classified as real or complex. Real roots are the x-values where the polynomial intersects the x-axis, while complex roots involve imaginary numbers and occur in conjugate pairs.
Finding Roots
Roots can be found using various methods such as factoring, synthetic division, or applying the quadratic formula for second-degree polynomials.
Multiplicity of Roots
Multiplicity refers to the number of times a particular root appears. A root with even multiplicity touches the x-axis, while a root with odd multiplicity crosses it.
Fundamental Theorem of Algebra
This theorem states that every non-constant polynomial has at least one complex root. Hence, a polynomial of degree n has exactly n roots in the complex number system.
Graphical Interpretation
The graph of a polynomial function shows its roots at the x-intercepts. The shape of the graph around the roots depends on their multiplicity.
Applications of Roots
Finding roots of polynomials is crucial in various fields like physics, engineering, and economics for solving equations related to real-world problems.
Elements of Galois theory
Elements of Galois theory
Introduction to Galois Theory
Galois theory establishes a connection between field theory and group theory, providing a powerful framework for understanding polynomial equations. It investigates the symmetries of roots of polynomials through group actions.
Field Extensions
A field extension is a bigger field containing a smaller field. Galois theory focuses on finite extensions of fields, particularly in algebraic contexts like solving polynomial equations.
Galois Groups
The Galois group of a field extension is the group of field automorphisms that fix the base field. It captures the symmetries of the roots of a polynomial and determines solvability by radicals.
Fundamental Theorem of Galois Theory
This theorem relates field extensions and subgroups of Galois groups. It states that there is a one-to-one correspondence between the intermediate fields of a field extension and the subgroups of its Galois group.
Applications of Galois Theory
Galois theory has profound implications in solving polynomial equations, especially determining whether a polynomial is solvable by radicals. It also plays a crucial role in number theory, algebraic geometry, and cryptography.
Finite fields - Wedderburn's theorem on finite division rings
Finite fields - Wedderburn's theorem on finite division rings
Introduction to Finite Fields
Finite fields, also known as Galois fields, are fields that contain a finite number of elements. The order of a finite field is a prime power, denoted by q. Elements in a finite field can be represented as polynomials over a finite field of prime order.
Definition of Division Rings
A division ring is a ring in which division is possible, except for division by zero. Every finite division ring is shown to be a field. The properties of division rings include associative multiplication, existence of multiplicative identity, and existence of inverses for non-zero elements.
Wedderburn's Theorem
Wedderburn's theorem states that every finite division ring is a field. This result is crucial for understanding the structure of finite division rings and establishes a strong link between division rings and fields.
Consequences of Wedderburn's Theorem
The theorem implies that division rings do not exhibit any exotic behavior in the finite case. It leads to the conclusion that finite division rings and finite fields can be studied under similar frameworks, emphasizing the importance of field theory.
Applications of Finite Fields
Finite fields have numerous applications in coding theory, cryptography, and combinatorial design. Their structure enables the establishment of error-detecting and error-correcting codes, as well as secure communication protocols.
Conclusion
The study of finite fields and the implications of Wedderburn's theorem are essential for advanced algebra, providing a deep understanding of algebraic structures and their applications in various mathematical and practical fields.
Solvability by radicals - Frobenius theorem - Integral Quaternions and the Four-Square theorem
Solvability by radicals, Frobenius theorem, Integral Quaternions and the Four-Square theorem
Solvability by radicals refers to the ability to express the roots of a polynomial equation using only arithmetic operations and radical (root) symbols.
The concept emerged during the Renaissance, with significant contributions from mathematicians like Tartaglia and Cardano.
A polynomial is solvable by radicals if its Galois group is a solvable group. This ties to the Abel-Ruffini theorem, indicating not all polynomial equations can be solved by radicals.
Quadratic equations are always solvable by radicals.
Cubic and quartic equations can also be solved by radicals under certain conditions.
The Frobenius theorem provides a criterion for the integrability of distributions, stating that a distribution is involutive if and only if the Lie bracket of any two vector fields in the distribution remains in the distribution.
Used in differential geometry, to study the existence of certain types of functions and the behavior of dynamical systems.
Understanding geodesic flows or Hamiltonian systems can provide insights into solving higher-order equations.
Integral quaternions are quaternions whose components are all integers. These numbers extend the concept of complex numbers and have applications in 3D computer graphics and physics.
Integral quaternions exhibit unique factorization properties similar to integers, leading to interesting algebraic structures.
The study of integral quaternions contributes to number theory and algebra, particularly in the context of solving equations in multiple dimensions.
The Four Square theorem, proven by Lagrange, states that every natural number can be represented as the sum of four integer squares.
If n is a natural number, then there exist integers a, b, c, d such that n = a^2 + b^2 + c^2 + d^2.
The theorem can be proven using properties of quaternions, specifically by considering the norm of quaternions.
This theorem has implications in number theory, combinatorics, and geometry.
