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Unformatted text preview: Math 4124 Wednesday, April 27 April 27, Ungraded Homework
Let k = Z/2Z, the ﬁeld with 2 elements. Prove that k[x]/(x5 + x2 + 1) is a ﬁeld with 32
First we show that x2 + x + 1 is the only irreducible polynomial of degree 2 in k[x]. Indeed
there are only 4 polynomials of degree 2, namely x2 + x + 1, x2 + 1, x2 + x, x2 . However the
last 3 are not irreducible, since x is a factor of the last 2, and if we plug in x = 1 in x2 + 1,
we get 0 and so x − 1 (= x + 1) is a factor of x2 + 1.
Next note that x5 + x2 + 1 has no degree 1 factor, because if we plug in x = 0 or 1, we
get 1. It follows that if x5 + x2 + 1 is not irreducible, it must have an irreducible degree
2 factor. Since the only irreducible degree two polynomial is x2 + x + 1, we deduce that
x5 + x2 + 1 = (x2 + x + 1) f for some f ∈ k[x]. but x5 + x2 + 1 = (x2 + x + 1)(x3 + x2 ) + 1,
which shows that x2 + x + 1 does not divide x5 + x2 + 1, and we have shown that x5 + x2 + 1
is irreducible. Since k[x] is a PID, it follows that k[x]/(x5 + x2 + 1) is a ﬁeld, with 25 = 32
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