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Prove that is a maximal ideal in Z5[x, y].

Chapter 14, Problem 43SE

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QUESTION:

Prove that \(\langle x, y \rangle\) is a maximal ideal in \(Z_5[x, y]\).

Questions & Answers

QUESTION:

Prove that \(\langle x, y \rangle\) is a maximal ideal in \(Z_5[x, y]\).

ANSWER:

Step 1 of 6

Solution: Let us consider the ring . We will propose to prove that the ideal  is a maximal ideal of the ring . Recall that,  is field of char 5 and if we assert that

then we must have  is a maximal ideal of the ring

. So it suffice to prove that

Claim:  is isomorphic to  as a ring.

Proof of the Claim: We will propose to show that

 

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