Tags: #todo #qual_algebra

# Algebra Fields Review

# Review

Tags: #field_theory #galois_theory #reading_notes

Exercises

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

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What is an algebraic closure of a field?

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What does it mean to be algebraically closed?

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What is an example of isomorphic but not equal fields?

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

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

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What are the following objects?

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

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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?

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

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