Study of cyclotomic extension of degree power of 2 and classification of radical extensions up to isomorphism
The primary focus of our conversation revolves around radical extensions. Schinzel’s theorem tells us that isomorphism classes of radical extensions come in two forms—one that’s well-known and another involving cyclotomic extensions of degree powers of 2. With that focus in mind, we will start our talk by delving into the interesting characteristics of quadratic cyclotomic extensions. This foundational knowledge will then help us explore cyclotomic extensions generated by primitive \((2^e)^{\text{th}}\) roots of unity, where \(e\) is a natural number. As we uncover these insights, we’ll wrap up by gaining a better understanding of the isomorphism classes of radical extensions and describing the moduli spaces that contain geometric information about these extensions.