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.
First, to show the polynomial x2 +x+1 is irreducible over Z2:
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 Z2 that is 0 and 1,
Put x=0 in the polynomial x2 +x+1:
Thus, the polynomial doesn’t have a zero.
Hence, the polynomial x2 +x+1 is irreducible over Z2.
Now, to show that the polynomial x2 +x+1 has a zero in an extension of Z2 :
First let if possible, α be a zero of x2 +x+1 in an extension of Z2
Now, doing long division:
Now, α will be a zero only if re...
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