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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.

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.

check_circleAnswer
Step 1

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,

Therefore,

First,

Put x=0 in the polynomial x2 +x+1:

02+0+1-1 in Z2
Now, put 1:
12+11 1 in Z2
help_outline

Image Transcriptionclose

02+0+1-1 in Z2 Now, put 1: 12+11 1 in Z2

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Step 2

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:

+(1a)
+1
+x
(x2
(1a)a
((1a) (1 a)a)
-ax)
+1
11a)a
help_outline

Image Transcriptionclose

+(1a) +1 +x (x2 (1a)a ((1a) (1 a)a) -ax) +1 11a)a

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Step 3

Now, α will be a zero only if re...

1+(1+a)a 1+a+a
=
= 2a2
0 (since, Z2 is a field of characteristic 2)
help_outline

Image Transcriptionclose

1+(1+a)a 1+a+a = = 2a2 0 (since, Z2 is a field of characteristic 2)

fullscreen

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