University of Nairobi Symmetric Polynomials Paper

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MAT 150C: MODERN ALGEBRA
Homework 8
Instructions. Please write the answer to each problem, including the computational ones, in connected
sentences and explain your work. Just the answer (correct or not) is not enough. Write your name in every
page and upload to Gradescope with the correct orientation. Make sure to indicate to Gradescope which
pages correspond to each problem. Finally, if you used another sources or discussed the problem
with classmates, be sure to acknowledge it in your homework.
1. Let f ∈ Q[x] be an irreducible polynomial of degree 4, and let K be the splitting field of f . Show that
[K : Q] ∈ {4, 8, 12, 24}.
√ √
√ √
2. Find α ∈ Q( 2, 5) such that Q( 2, 5) = Q(α). Note: You need to prove that the two extensions
are equal.
3. Express each of the following symmetric polynomials as a polynomial in the elementary symmetric
polynomials.
(a) u31 u2 + u32 u1 + u31 u3 + u33 u1 + u32 u3 + u33 u2 .
(b) (u1 − u2 )2 .
(c) u31 + u32 + · · · + u3n . (Hint: Do the cases n = 2, 3 separately)
4. Let F be a field of positive characteristic p > 0 and let F be an algebraic closure of F. Let n > 0 and
let q := pn . Show that the set
Fq := {α ∈ F : αq = α}
is a subfield of F, and that Fq has at most q elements..1
1
It can be shown that this set has exactly q elements and, moreover, that up to isomorphism it is the unique field with q
elements. It is usually denoted by Fq .
1

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