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# Fields - Cardinality and Splitting Fields.

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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?

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https://brainmass.com/math/algebra/fields-cardinality-splitting-fields-467403

#### 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.

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