# Fields - Cardinality and Splitting Fields.

We know that F_2[x]/(x^2 + x + 1) is a field of cardinality 4; call it F_4.

Find an irreducible quadratic f(y) [element of] F_4[y].

What is the cardinality of the finite field F_4[y]/(f(y))?

This field is isomorphic to the splitting field for x^2^n - x over F_2[x]. What is the appropriate n in this case?

Show that x^4 + x^3 + x^2 + x + 1 [element of] F_2[x] is irreducible.

Find the cardinality of the field F_2[x]/(x^4 + x^3 + x^2 + x + 1).

This field is isomorphic to the splitting field for x^2^n - x over F_2[x]. What is the appropriate n in this case?

Please see the attachment for complete equations.

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

1. The elements of the finite field are 0, 1, , and , where An irreducible quadratic polynomial over is The cardinality of is is isomorphic to the splitting field of over , i.e. we have

2. To see that the polynomial is irreducible, first we show that it has no roots in . Since we have we see that neither 0 nor 1 is a root. Next we must show that is not the product of two quadratic polynomials in . Suppose it were. Then we would have whence The last of these equations implies and implies that one of a and c is 0 and the other is 1, which in turn implies , whence the second equation becomes But since we have yielding a contradiction. Thus is not the product of two quadratic polynomials, so it is irreducible.

The cardinality of is K is isomorphic to the splitting field of over , i.e. we have

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