Find a minimal polynomial of α when α is an irrational number satisfying α 3 + 3 α 2 − 2 = 0.

Question

Find a minimal polynomial of   α when   α is an irrational number satisfying       α    3    +  3      α    2    −  2  =  0.

Govorei9b · Accepted Answer

The polynomial       X    3    +  3      X    2    −  2 factors as      X    3    +  3      X    2    −  2  =  (  X  +  1  )  (      X    2    +  2  X  −  2  )  .The quadratic factor is irreducible (over             Q      ) because it has no roots (in             Q      ).Since   α is a root of       X    3    +  3      X    2    −  2, it is a root of either   X  +  1 or of       X    2    −  2  X  −  2. Because   α is irrational, it is not equal to   −  1, so it is a root of       X    2    +  2  X  −  2.So,       X    2    +  2  X  −  2 is an irrediducible, monic polynomial with   α as a root, so it is the minimal polynomial of   α.

Find a minimal polynomial of α when α is an irrational

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