Let $K$ be a field and $f(x)$ be an irreducible polynomial over $K$. Suppose, $f(x)$ has degree at least $2$. Is it possible that if $a,b$ are two roots of $f(x)$ with $a\neq b$, then $K(a)=K(b)$. Note I need equality, not isomorphism.
Can two different roots of an irreducible polynomial generate the same extension?
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field-theory
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1As Qiaochu notes in his comment to my answer, you really need to specify that you are working inside a fixed algebraic closure of $K$ for the question of equality of the two extensions to make sense. – 2010-12-12
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0This is always true if the degree of $f$ is exactly 2. – 2012-02-26