Are Irreducible Polynomials of Degree 4 Always Reducible?

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Proposition. A polynomial of degree 2 or 3 over a field F is reducible iff it has a root in F.

Tell me if I'm on the right track... I see that x^4 + 3x^2 + 2 is reducible (x^2+1)(x^2+2) but has no roots in Q.

This serves as a counterexample to the proposition if states for polynomial up to degree 4?
 
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Yes, that serves as a counter example for polynomials of degree 4. And that's why your proposition only says "degree 2 or 3".

If a polynomial of degree three is "reducible", then it must be "reduced" to two linear factors. And those two linear factors will give roots. If a polynomial of degree three is "reducible" then it is "reduced" to either a product of three linear factors or a product of a linear factor and an irreducible quadratic. In either case, you have at least one linear factor that gives a root.
 
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