Mathematics 894 Problem Set 10

Text: Abstract Algebra, Third Edition, by Dummit and Foote

Suggested reading: Sections 13.4, 13.5, 13.6. Also Theorem 2.8 page 233 of Lang (third edition). This permits an easy proof of the uniqueness of the algebraic closure of a field (which Dummit and Foote don’t quite do), and also of the existence and uniqueness of the splitting field of a polynomial, replacing Theorem 27 page 541 of Dummit and Foote.

Questions due next Friday (March 28, 08)

1. Page 545: 5

2. Let n be an integer greater than 1, and let d ≥ 1 be a divisor of n. Let p be a prime number. Use Problem 1 to Prove that every irreducible polynomial of degree d over 𝔽_{p} = ℤ/pℤ divides X^{p^{n}} − X.

3. Factor X^{16} −X ∈ 𝔽_{2}[X] as the product of irreducible factors. Use Problem 2 to list all the irreducible polynomials of degree 1,2, and 4 in 𝔽_{2}[X].

4. Regard each field 𝔽_{p^{n}} as a subfield of the algebraic closure \overline{𝔽}_{p} of 𝔽_{p}, so that it makes sense to talk about inclusion of such fields. Prove that 𝔽_{p^{d}} ⊆ 𝔽_{p^{n}} if and only if d|n.

5. Page 551: 5

6. Page 555: 4,5,6