How does cancellation work in polynomial quotient rings?Z[x](2x−6,6x−15)Can I just automatically say thatZ[x](2x−6,6x−15)=∼Z[x](x−3,6x−15)by just dividng the first polynomial by 2?

Step 1Yes, we can cancel 2. Call the ideal I.(2)+I=(2,2x−6,6x−15)=(2,15)=(1)&nbsp;so by Euclid's LemmaI⊇2(x!−!3)⇒I⊇(x!−!3),SoI=(x!−!3,I&nbsp;mod&nbsp;x!−!3)≠!(x!−!3,3)=(x,3)Simpler(2x−6,6x−15)=(2x−6,3)=(x,3)using(a,b)=(a,b&nbsp;mod&nbsp;a)

Expert answers to "How does cancellation work in polynomial quotient rings?Z[x](2x−6,6x−15)Can..."

College
Algebra
Abstract algebra

Saige Shannon

Answered question

2022-04-22

How does cancellation work in polynomial quotient rings?

\frac{Z [x]}{(2 x - 6, 6 x - 15)}

Can I just automatically say that

\frac{Z [x]}{(2 x - 6, 6 x - 15)} \tilde{=} \frac{Z [x]}{(x - 3, 6 x - 15)}

by just dividng the first polynomial by 2?

Answer & Explanation

pecanjima9vo

Beginner2022-04-23Added 11 answers

Step 1
Yes, we can cancel 2. Call the ideal I.
$(2) + I = (2, 2 x - 6, 6 x - 15) = (2, 15) = (1)$ so by Euclid's Lemma
$I \supseteq 2 (x! -! 3) \Rightarrow I \supseteq (x! -! 3),$
So
$I = (x! -! 3, I mod x! -! 3) \neq! (x! -! 3, 3) = (x, 3)$
Simpler
$(2 x - 6, 6 x - 15) = (2 x - 6, 3) = (x, 3)$
using
$(a, b) = (a, b mod a)$

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Abstract algebra

How to find out the mirror image of a point?
Generators of a free group
If G is a free group generated by n elements, is it possible to find an isomorphism of G with a free group generated by n-1 (or any fewer number) of elements?
How many 3/4 Are in 1
Convert 10 meters to feet. Round your answer to the nearest tenth
6. Reduce the following matrix to reduced row echelon form:
$C equals open parentheses table row 1 cell negative 1 end cell cell 1 half end cell cell 1 half end cell row 0 1 cell 1 half end cell cell negative 1 half end cell row 0 0 1 0 row 0 0 0 0 end table close parentheses$
Let v be a vector over a field F with zero vector 0 and let s,T be a substance of V .then which of the following statements are false
Describe Aut(Zp), the automorphism group of the cyclic group Zp where p is prime. In particular find the order of this group. (Hint: A generator must map to another generator)

Didn't find what you were looking for?