Show steps -- Write each of the following polynomials as a product of irreducibl

Question

  Show steps -- Write each of the following polynomials as a product of irreducible polynomials over the given field: X5 + X + 1 over Z2, X4 + 4 over Q.

Explanation / Answer

(b) Firstly, over Z2, it helps knowing thepolynomials of degree 2 that is irreducible. It is easy to see thatthere is only 1 polynomial of degree 2 that is irreducible, namely,g(X)=X2 + X + 1. Similarly, the only irreduciblepolynomials of degree 3 are X3 + X2 + 1 andX3 + X + 1. Now, note that since the given polynomialhas no roots in Z2 (since plugging 0 or 1 does not givezero), => that if the given polynomial is not irreducible thenit must be factored into a polynomial of degree 2 and a polynomialof degree 3. Now it is easy to check that    X5 + X + 1=(X2 + X +1)(X3 + X2 + 1). (c) This part becomes apparent after this bit of algebra: X4 + 4 = (X2 + 2)2 -4X2 = (X2 + 2)2 - (2X)2. This is of the form (a2 - b2 ) and soadmits the factors (a+b) and (a-b), so we have X4 + 4 = (X2 -2X+2)(X2 +2X+2). Itis easy to also see that both these are irreducible over Q, andthat is the required factorization.

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