Answer to Question #105108 in Abstract Algebra for Sourav Mondal

Question #105108
Check whether or not <(x+1)²> is a maximal ideal of Z[x].
1
Expert's answer
2020-03-16T13:35:27-0400

The maximal ideals of "Z[x]" are of the form "(p,f(x))" where "p" is a prime number and "f(x)" is a polynomial in "Z[x]" which is irreducible modulo "p".

To prove this let "M" M be a maximal ideal of "Z[x]" . Assume that "M\\cap Z[x]\\neq(0)" . As "Z\/(M\\cap Z)" injects into "Z[x]\/M" , we conclude that "Z\/(M\\cap Z)" is a domain, so "M\\cap Z" is a prime ideal of "Z" so "M\\cap Z=(p)" , where "p" is a prime number.


In our case

"n=1,\\\\\nf(x)=<(x+1)^2>=(x+1,x+1)=x+1 ."

"n" is not a prime number, then "<(x+1)^2>" is not a maximal ideal of "Z[x]" .


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