View source for Ring Theory/Ring extensions

{{definition|ring extension|Whenever <math>S</math> is a ring and <math>R \subset S</math> is a subring of <math>S</math>, we say that <math>S</math> is a '''ring extension''' of <math>R</math> and write <math>S/R</math>.}}

Note that if <math>S/R</math> is a ring extension, then <math>S[x]/R[x]</math> is a ring extension; indeed, the set <math>S[x]</math> is the set of all polynomials with coefficients in <math>S</math>, the set <math>R[x]</math> is the set of all polynomials with coefficients in <math>R</math>, and <math>R[x]</math> is a subring of <math>S[x]</math>.

{{proposition|existence of splitting ring|Let <math>R</math> be a ring, and let <math>p \in R[x]</math> be a polynomial over <math>R</math>. Then there exists a ring extension <math>S/R</math> such that in <math>\overline S</math>, <math>p</math> decomposes into linear factors, that is,
:<math>p(x) = (x - \lambda_1) \cdots (x - \lambda_n)</math> for certain <math>\lambda_1, \ldots, \lambda_n \in \overline S</math>.}}

{{proof|We prove the theorem by induction on the degree of <math>p</math>. Suppose first that <math>p</math> can be decomposed into two polynomials }}
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