Question

Prove that \frac{\mathbb{R}[x]}{\left \langle x^{2}+1 \right \rangle}\simeq \mathbb{C}     as rings.

11 Mar 2024
Answer :
Word Count : 366
We want to prove that the quotient ring \[ \frac{\mathbb{R}[x]}{\langle x^2 + 1 \rangle} \simeq \mathbb{C} \] as rings. ### Step 1: Understanding the Quotient Ring The ring \(\mathbb{R}[x]\) consists of all polynomials with real coefficients. The ideal \(\langle x^2 + 1 \rangle\) is generated by the polynomial \(x^2 + 1\), meaning that in the quotient ring, we consider two polynomials ____ __________ _________ __________ ________ ______ ________ ________ _______ ________.
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