Irreducible Polynomial/Examples/x^2 - 2 in Reals
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Examples of Irreducible Polynomials
Consider the polynomial:
- $\map P x = x^2 - 2$
over the ring of polynomials $\R \sqbrk X$ over the real numbers.
Then $\map P x$ is not irreducible, as from Difference of Two Squares:
- $x^2 - 2 \equiv \paren {x + \sqrt 2} \paren {x - \sqrt 2} $
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: Polynomials and Euclidean Rings: $\S 29$. Irreducible elements: Example $58$