Math 150B Homework 4
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Math 150B Homework 4
1) True or false? Give a one sentence justification of each one.
a) Every field is a UFD.
b) Every field is a PID.
c) Every PID is a UFD.
d) Every UFD is a PID.
e) 2[x] is a UFD.
f) Any two irreducibles in any UFD are associates.
g) If D is a PID, then D[x] is a PID.
h) If D is a UFD, then D[x] is a UFD.
i) A UFD has no zero divisors.
2) Use a Euclidean algorithm in 2[i] to find a gcd of 16 + 7i and 10 _ 5i in 2[i].
3) Prove that if p is an irreducible in a UFD, then p is prime.
4) Let R be any ring. The decending chain condition (DCC) for ideals holds in R if every strictly decreasing sequence N1 5 N2 5 N3 5 ... of ideals in R is of finite length. The minimum condition (mC) for ideals holds in R if given any set S of ideals in R, there is an ideal in S that does not properly contain any other ideal in S.
Show that in any ring the conditions DCC and mC are equivalent.
5) Let D be a Euclidean domain and let v be a Euclidean norm on D. Show that if a ﹐ b are associates in D, then v(a) = v(b).
6) Show that every field is a Euclidean domain.
7) Let D be a UFD. An element c e D is a least common multiple (lcm) of two elements a and b in D if alc﹐ blc, and if c divides every element of D that is divisible by both a and b. Show that every two nonzero elements in a Euclidean domain D have an lcm in D. (Hint: show that all common multiples of a ﹐ b form an ideal in D)
8) Show that for p prime, the polynomial xp + a is not irreducible for any a e 2p .
9) Find the number of irreducible quadratic polynomials in 2p[x], where p is a prime.
10) How difficult was this homework? How long did it take?
2022-02-08