Download Problem set 5

MATH 210
Problem Set 5
Due: February 12, 2016
Exercises
1. (a) Find all roots of f (x) = x3 − x2 − 5x − 3 in Q.
(b) Find all roots of g(x) = x3 − 3x2 − 12x + 36 in R.
(c) Find all roots of h(x) = x3 + 1 in C.
2. Determine whether the following polynomials are irreducible in their respective fields of definition.
(a) f (x) = x4 + 2x3 + 3x2 + 2x + 1 in Q[x],
(b) g(x) = 2x3 + 3x2 + x + 1 in Q[x],
(c) h(x) = 2x7 + 5x5 + 15x4 + 10x3 + 35x2 + 25x + 45 in Q[x],
(d) x2 + 1 in C[x].
3. (a) Show that x2 + 1 ∈ R[x] is irreducible, and conclude that R[x]/(x2 + 1)R[x] is a field (see problem
5). Show that there is an element f (x) ∈ R[x]/(x2 + 1)R[x] satisfying (f (x))2 = −1.
(b) Explain how a complete system of residues for R[x]/(x2 + 1)R[x] is given by {a + bx : a, b ∈ R}.
(Does this look familiar? Hm...)
Problems
4. (a) Show that if f (x) ∈ R[x] has a root α ∈ C, then its complex conjugate α is also a root, i.e.,
f (α) = 0 implies f (α) = 0.
(b) Show that if f (x) ∈ R[x] is irreducible, then it is linear, or f (x) = ax2 + bx + c for some a, b, c ∈ R,
with b2 − 4ac < 0.
5. (a) Let F be a field. Show that f (x) ∈ F [x] is irreducible if and only if every non-zero element in
F [x]/f (x)F [x] is a unit.
(b) Show that strange things happen when not working with a field F , but a ring R instead. In
particular, show that if R = Z/6Z and g(x) = 3x + 1 ∈ R[x], then R[x]/(g(x)R[x]) does not
contain a root of g(x). More generally, show that no ring containing R can contain a root of g(x).
Food for thought
Consider a pair of (ordinary) 6 sided dice, each labeled 1 through 6. Consider another pair of (non-ordinary)
dice, one of which is labeled 1, 2, 2, 3, 3, 4, and the other 1, 3, 4, 5, 6, 8.
One can show the probability of rolling a number n with the first pair of dice is equal to the probability
of rolling n with the second pair of dice, e.g., the probability of rolling a 2 is 1/36 with both pairs, and rolling
a 5 has probability 4/36, etc.... This isn’t terribly difficult to show, as one can simply check all possible
rolls with both pairs and see the probabilities are equal. What’s special is that these are the only two pairs
of dice with these probabilities. Use unique factorization in Z[x] to show that no other pair of dice (with
natural numbers on each face) have the same probability for each n ∈ N.
Practice problems
Shifrin section 3.1: 7, 9, 10, 11, 12, 13, 14. Section 3.3: 2, 3, 4, 5, 6.
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