Question: is polynomial irreducible in let be the ideal of...
Question details
Is polynomial 
irreducible in ![\mathbb{Z}_5[x]](https://homework-api-assets-production.s3.ap-southeast-2.amazonaws.com/uploads/store/590001646/16040306457140cac0854bb565ab726a705153c5a2.gif)
?
Let ![[1+x+x^2 ]](https://homework-api-assets-production.s3.ap-southeast-2.amazonaws.com/uploads/store/590001646/1604030646737eb11e3a7550db682f46af7c3feb16.gif)
be the ideal of ![\mathbb{Z}[x]](https://homework-api-assets-production.s3.ap-southeast-2.amazonaws.com/uploads/store/590001646/1604030647b788537b70e30b4fd382f4de8662b87c.gif)
generated by 
. Is the quotient ring ![\mathbb{Z}/[1+x+x^2]](https://homework-api-assets-production.s3.ap-southeast-2.amazonaws.com/uploads/store/590001646/16040306484b6f600840c9422239da8e912e13119f.gif)
a field? How many elements does the quotient ring ring ![\mathbb{Z}_5[x]/[1+x+x^2]](https://homework-api-assets-production.s3.ap-southeast-2.amazonaws.com/uploads/store/590001646/16040306498e7c2c7fae61ba2d5401d31e2b4a6b92.gif)
have?
Solution by an expert tutor
