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Fall 2007
Elementary Abstract Algebra I Practise Final
Name:
1 2 3 4 5 6 7 8 TOT.
FINAL EXAM: 15:15โ17:05 THURSDAY OF FINALS WEEK.
The real final will look roughly like this, probably slightly shorter questions, but similar topics. Sections to revise: Chapter 1 (all), Section 2.1, Chapter 4 (all), Section 5.1, Definition 5.2.1 (why is it automatic that ฯ(1) = 1 if ฯ is one-to-one and onto?). Then go over all homeworks and make sure you understand how to do them with hindsight! Revising is a key part of learning mathematics: lots of things you didnโt completely understand the first time round should be easier when you go back over it again!!
- Let F be a field and f (x) โ F [x] be an irreducible polynomial. Prove carefully that f (x)|a(x)b(x) implies either f (x)|a(x) or f (x)|b(x).
- (a) For which values of a = 1, 2 , 3 , 4 is Z 5 [x]/ใx^2 + x + aใ a field? Explain.
(b) For which values of k = 2, 3 , 4 , 5 is Zk[x]/ใx^2 + x + 1ใ a field? Explain (and be extra careful with k = 4).
- (a) Let F be a field and f (x) โ F [x] be a polynomial. Prove the remainder theorem: โFor a โ F , (x โ a) is a linear factor of f (x) if and only if f (a) = 0.โ
(b) Prove that f (x) = x^3 + 3x^2 + 2x + 1 is irreducible over Q. (Hint: draw a rough sketch of the graph by plotting points at x = โ 3 , โ 2 , โ1 and 0 and use a recent theorem about rational roots...)
- (a) Prove for any x โ Z that x^5 โก x (mod 5).
(b) Find the multiplicative inverse of x in Z 5 [x]/ใx^2007 + 2ใ.
(c) Is the ring Z 5 [x]/ใx^2007 + 2ใ a field?
- For each of the following, decide whether the function is well-defined, and if so determine whether it is one-to-one and/or onto.
(a) f : C โ C, x + iy 7 โ 2 y + i(
โ 3 2 x^ โ^ y).
(b) f : Z 4 โ Z 8 , [x] 4 7 โ [3x] 8.
(c) f : R ร R โ C, (x, y) 7 โ x + iy.
(d) f : R โ C, x 7 โ x^2 + ix.
(e) f : C ร C โ C ร C, (x, y) 7 โ (x^2 , x + y).
- Here are five rings. Work out which of them are isomorphic to each other. Explain! R 1 = Z 2 [x]/ใx^3 + xใ. R 2 = Z 8. R 3 = Z 2 ร Z 4. R 4 = Z 2 [x]/ใx^2 + x + 1ใ. R 5 = Z 2 ร Z 2 ร Z 2.
