This week, we are covering the following sections:

• §1.1 Rings
• §1.2 Fields
• §1.3 The characteristic of a field
• §1.4 Review of polynomial rings
• §1.5 Factoring polynomials
• §1.6 Extensions
• §1.7 The subring generated by a subset
• §1.8 The subfield generated by a subset
• §1.9 Construction of some extensions
• §1.10 Stem fields

MATH3303 review questions:

1. Let \(R\) be a commutative ring and \(I\subseteq R\) an ideal. Prove that \(R/I\) is an integral domain if and only if \(I\) is prime. Prove that \(R/I\) is a field if and only if \(I\) is maximal. Deduce that maximal ideals are prime.
2. Let \(R\) be a Euclidean domain. Prove that \(R\) is a principal ideal domain, and also a unique factorisation domain.
3. Let \(R\) be a unique factorisation domain. Prove that \(a \in R\) is prime if and only if it is irreducible.

Exercises 1—3, 5—7 from Chapter 1. After completing Exercise 3, solve review exercise A-4 (b).

Additional exercises:

1. Prove that if a field has nonzero characteristic \(n\), then \(n\) is a prime number.
2. Find an infinite field with nonzero charactersitic.
3. Let \(F\) be a field with characteristic \(p\). Milne notes that \(a \mapsto a^p\) is an automorphism, which is called the Frobenius automorphism. Explain why the Frobenius automorphism is trivial for \(F = \mathbb{F}_p\), and use an explicit construction of \(\mathbb{F}_4\) to show its Frobenius automorphism is nontrivial.
4. In a field \(F\), the equality \(a^4 = a\) is satisfied for all \(a \in F\). Find the characteristic of \(F\).
5. Show that \(X^4 + 2X^2 + 9 \in \mathbb{Q}[X]\) has no rational roots, but is reducible in \(\mathbb{Q}[X]\).
6. Make a list of all irreducible polynomials in \(\mathbb{F}_2[X]\) of degree less than or equal to 5.
7. Factorise the following as a product of irreducible polynomials:
   1. \(X^4 + 64\) in \(\mathbb{Q}[X]\).
   2. \(X^4 + 1\) in \(\mathbb{R}[X]\).
   3. \(X^7 + 1\) in \(\mathbb{F}_2[X]\).
   4. \(X^4 + 2\) in \(\mathbb{F}_{13}[X]\).
   5. \(X^3 - 2\) in \(\mathbb{Q}[X]\).
   6. \(X^6 + 27\) in \(\mathbb{R}[X]\).
   7. \(X^3 + 2\) in \(\mathbb{F}_3[X]\).
   8. \(X^4 + 2\) in \(\mathbb{F}_3[X]\).
8. Use Gauss's Lemma to show that the following polynomials are irreducible in \(\mathbb{Q}[X]\):
   1. \(X^4+8\).
   2. \(X^4-5X^2+2\).
   3. \(X^4-4X^3+12X-16X+8\).
9. Prove \(p^{n-1} X^n + p X + 1\) is irreducible over \(\mathbb{Q}\) for every prime \(p\) and positive integer \(n\).
10. Show that over every field there exists infinitely many irreducible polynomials.