Question

Show that x^2 + x + 1 is irreducible over Z_2 and has a zero in some extension field of Z_2 that is a simple extension. This is a problem for abstract algebra that I am struggling with.

Step 1

First, to show the polynomial *x*^{2} +*x*+1 is irreducible over Z_{2}:

Here, recall that a polynomial of degree 2 or 3 is irreducible if and only if it does not have a zero.

Now, since there are only two elements in Z_{2} that is 0 and 1,

Therefore,

First,

Put *x*=0 in the polynomial* x*^{2} +*x*+1:

Step 2

Thus, the polynomial doesn’t have a zero.

Hence, the polynomial *x*^{2} +*x*+1 is irreducible over Z_{2}.

Now, to show that the polynomial *x*^{2} +*x*+1 has a zero in an extension of Z_{2 }:

First let if possible, α be a zero of *x*^{2} +*x*+1 in an extension of Z_{2}

Now, doing long division:

Step 3

Now, α will be a zero only if re...

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