Tags: #todo #qualifying_exam #qual_algebra

Algebra Fields Review

Review

Tags: #field_theory #galois_theory #qualifying_exam #reading_notes

Exercises

How do you construct finite fields with \(p^n\) elements?

What is an algebraic closure of a field?

What does it mean to be algebraically closed?

What is an example of isomorphic but not equal fields?

Why is every finite extension algebraic? Why is the degree of an extension given by joining algebraic elements always finite?

Why is \([K\(alpha): K]\) equal to the degree of the minimal polynomial of \(\alpha\) when it is algebraic?

What are the following objects?

• \(K(x)\)
• \(K[x]\)
• \(K( \alpha)\)
• \(K[ \alpha]\)

What does it mean for an element to be algebraic or transcendental? Which is \(x\in {\mathbf{C}}(x)\)? What is an example of a clearly non-algebraic element of a field?

What is the degree of a field extension? What does it mean to be a finite extension?