MATH:5010 - Spring 2017

Homework Assignment 8

Reading (from Dummit and Foote):

Read Sections 13.4, 13.5. Also read Sections V.2, V.3 from Lang's "Algebra" book (these sections can be found on ICON).

Please look through the sections for EACH lecture before the lecture. See the Lecture Schedule for the sections that will be covered.

Then after EACH lecture, review your lecture notes and thoroughly read the sections covered in class.

Part A Problems (to be handed in on Tuesday, March 28, at the beginning of the discussion section):

Section 13.2: # 19, 21.
Section 13.4: # 2, 3.

Part B Problems (to be handed in on Friday, March 31, at the beginning of the lecture):

If you hand in ALL Part B Problems together with the Part A Problems already on Tuesday, March 28, 2 extra points will be added to your homework score.

Section 13.2: # 18, 20.
IMPORTANT for #18: Assume that t does not lie in k, i.e. at least one of P(x) or Q(x) is not a constant polynomial in k[x].
IMPORTANT for #20: Use the notation introduced in #19. The polynomials to be obtained in the last sentence of #20 should be over the rational numbers.
Section 13.4: # 6.
IMPORTANT:
- The assumption that K₁ and K₂ are splitting fields means that each of them is a splitting field of a family of polynomials in F[x] of degrees ≥ 1. Since we assume K₁ and K₂ to be finite over F, we can take these families to be finite. Hence, by taking the product of the polynomials in each of these families we get that K₁ (resp. K₂) is the splitting field of a single polynomial f₁(x) (resp. f₂(x)) in F[x] of degree ≥ 1.
- Note that we proved a stronger version of #5 in the lecture on Wednesday, 03/22. So you can assume #5 without proof.

Additional Practice Problems (you have to know how to do these, but you do NOT have to hand these in):

Section 13.2: # 3, 4, 5, 6, 7, 10, 11, 12, 13.
Section 13.4: # 1, 4.

Frauke Bleher
Mar 22 2017