# 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 Preview

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 ...

#### Solution Summary

We solve two problems involving polynomials over finite fields. Response includes Microsoft Word attachment.