Answer to Question #194922 in Abstract Algebra for Gaurav Panchal

Question

check whether or not Q[x]/<8x^3+6x^2-9x+24> is a feild or not

Answer 1

Question:- check whether or not Q[x]/<8x³+6x²-9x+24> is a field or not?

solution:- This type question solve with

Eisenstein Criterion rule.

first of all , definition of Eisenstein Criterion rule.

comes on question again,

when we check Q[x]/<polynomial> is field or not, then we check, polynomial is irreducible or not.

for check of irreducibility, we use Eisenstein Criterion rule .

Now in this question we will check polynomial is irreducible or not.

polynomial is given as:-

<8x³+6x²-9x+24>

according to Eisenstein Criterion rule,

here a prime p=3 exist.

such that

(1)3 does not divide 8

(2)3 divide all other coefficient 6,9 and 24

(3)3² does not divide 24

so polynomial is irreducible.

hence Q[x]/<8x³+6x²-9x+24> is a feild.