Math 366–001 HW 8, Spring 2015
This assignment is due Monday, May 4. Feel free to work together, but be sure to write
up your own solutions. As for writing it up, please write legibly on your own paper, including
as much justification as seems necessary to get the point across.
All problems are worth 2 points.
1. Write the splitting field of x4 + x2 + 1 = (x2 + x + 1)(x2 − x + 1) over Q in the form
Q(a) for some number a.
2. Let E be a finite extension of R. Use the fact that C is algebraically closed (meaning
that all algebraic extensions of C have degree 1) to prove that E = C or E = R.
√ √ √
3. Assume that Q( 2, 3 2, 4 2, . . .) is an algebraic extension of Q. (This is true, but you
don’t need to show it.) Show that it is not a finite extension (so finite implies algebraic,
but not vice versa!).
4. Find [GF (1024) : GF (2)]. (Corollary 1 of Chapter 22 could help!)
5. Find [GF (1024) : GF (32)].
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