Word Count : 548

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To prove that \(\frac{\mathbb{R}[x]}{\langle x^2 + 1 \rangle} \simeq \mathbb{C}\) as rings, we will follow these steps: ### Step 1: Understand the quotient ring \(\frac{\mathbb{R}[x]}{\langle x^2 + 1 \rangle}\) This quotient ring is the ring of polynomials in \(x\) with real coefficients, modulo the ideal \(\langle x^2 + 1 \rangle\), which is generated by the polynomial \(x^2 + 1\). The elements of the quotient ring are equivalence classes of polynomials, where two polynomials are equivalent if their difference is a multiple of \(x^2 + 1\). ### Step 2: Interpret the structure of the quotient ring We can think of the quotient ring \(\frac{\mathbb{R}[x]}{\langle x^2 + 1 \rangle}\) as consisting of polynomials in \(x\) with real coefficients, but with the relation \(x^2 = -1\) imposed. So, in this ring, the variable \(x\) behaves just like the imaginary unit ________ _______ ____ _________ ____ _________ _______ _________ _____ _____ _________ ________.
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