The lecturer said I was reading in the wrong place and that we aren't doing any work on quadratic integer rings - and that to determine that 3 is irreducible in Q[i sqrt(5)] requires a "field norm
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First, there's a typo somewhere. I think that the polynomial should be x3 - nx + 2.
Next, for the problem to make sense, the underlying ring must be Q or Z. If the problem (or the previous text) doesn't make that explicit, then it ought to.
To solve the problem, notice that if a cubic is reducible, then one of the factors must be linear, and so the polynomial must have a (rational or integer) root. The constant term is 2, so the rational root test says that the only possible roots are ±1 and ±2. Plug those values of x into the equation x3 - nx + 2 = 0 and you can read off the possible values of n.
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Next, for the problem to make sense, the underlying ring must be Q or Z. If the problem (or the previous text) doesn't make that explicit, then it ought to.
To solve the problem, notice that if a cubic is reducible, then one of the factors must be linear, and so the polynomial must have a (rational or integer) root. The constant term is 2, so the rational root test says that the only possible roots are ±1 and ±2. Plug those values of x into the equation x3 - nx + 2 = 0 and you can read off the possible values of n.
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