113 class problems: irreducible elements and unique
TRANSCRIPT
113 Class Problems: Irreducible Elements and Unique
Factorization Domains
1. Is the polynomial 2x2 − 4 irreducible in C[x]? How about in R[x], Q[x] or Z[x]?Solutions:
2. If a polynomial f(x) ∈ Q[x] has no roots in Q must it be irreducible in Q[x]?
Solutions:
EC c Cx t
UT 6 ReducibleCx 2 2 4 2 x Tz x fr6 RE e IRC
in Ecsc
ReducibleIRCx 2 2 4 Z x 152 Cx T2 in RexCx 2 2 4 axe b Coxed a b c d C
I 9 3 Vz Vz Vz C GutradictionIrreducible in Cx
26 3 zc.cz z Reducible in ZCxZC y Eca
No x2 Cx2 is reducible ai Cx but has
no roots in
3. Consider the subring Z[√−5] ⊂ C
(a) Prove that Z[√−5] = {a+ b
√−5|a, b ∈ Z}
(b) If a + b√−5 ∈ Z[
√−5] is non-zero, what is the minimum possible value of |a +
b√−5|2, the square of the absolute value?
(c) Using part (b) determine Z[√−5]∗, the units in Z[
√−5].
(d) Prove that 2, 3, 1 +√−5, 1−
√−5 are non-associated elements of Z[
√−5].
(e) Prove that 2, 3, 1 +√−5, 1−
√−5 are irreducible elements of Z[
√−5].
(f) Prove that Z[√−5] is not a UFD.
Solutions:
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a
Fs S E Z Ects atb Fs a bee
I la tbFsl2 art 5b2 a b C Z
lax bfs 12 if a b C Z Min value is when aand b O
I L D E Z Cfs late 113T I
2 3 1 121211312 1 1 12 11312 1 I
Cfs t
d 2 3 I f s l f s are pairwise non associated as
2 Cfs tft and non is a negative of another
2 BE Ectse
2 2 3 22 12121 312 I E 1,2 4
a Sbl 2 has no integer solutions la I _I n 4
1 12 1 c Effs I 12 4 11312 1 BE ECFSI
z irreducible
Same logic shows 3 It F s l Fs are all irreducible
G Z 3 I Fsk yestwo non associatedirreducible Factorization
Cfs not a VFD