This module explores the power of symmetry to present key results in Group Theory, Galois Theory, and Coding Theory. It begins with fundamental group-theoretical methods before uncovering the deep connections between group actions and roots of polynomials, leading to the fundamental theorems of Galois theory. A highlight Is the proof of the insolubility by radicals of the general quintic polynomial. The course concludes by utilizing the theory of finite fields and vector spaces to coding theory, highlighting both theoretical insights and practical applications. Blending elegant algebraic structures with real-world relevance, this module provides a thorough and engaging overview of these interconnected topics, and highlights abstract mathematics along with its real-world applications.
Learning Outcomes
By the end of the module students should be able to:
Understand and apply fundamental concepts of group theory, including group actions, Sylow theorems and solubility.
Understand and apply the theory of field extensions, including algebraic and transcendental extensions, and analyze their properties.
Comprehend and apply the theory of finite fields.
Understand and prove key results in Galois theory, including their repercussions in solving polynomial equations which relates soluble groups to bring to finding roots of polynomials.
Comprehend the principles of coding theory and its applications in error detection and correction.
Evaluate the interplay between abstract algebraic structures and practical coding methods.
