Galois extensions for cyclic group:Z2: Difference between revisions
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In this case, the quadratic extension can be obtained by adjoining a polynomial of the form <math>x^2 - a</math> where <math>a</math> is a nonzero element of the field that is not a square. For prime fields, such an element is termed a quadratic nonresidue. | In this case, the quadratic extension can be obtained by adjoining a polynomial of the form <math>x^2 - a</math> where <math>a</math> is a nonzero element of the field that is not a square. For prime fields, such an element is termed a quadratic nonresidue. | ||
===Case where the field has characteristic two=== | |||
This case is a little trickier, since no polynomial of the form <math>x^2 - a</math> is irreducible. Rather, an irreducible polynomial is of the form <math>x^2 + x + a</math>. | |||
Revision as of 21:35, 14 April 2009
Given a Galois extension of fields, the Galois group for the extension is the cyclic group of order two if and only if the field extension is a quadratic extension.
Over a finite field
Over a finite field, there is exactly one quadratic extension.
Case where the field does not have characteristic two
In this case, the quadratic extension can be obtained by adjoining a polynomial of the form where is a nonzero element of the field that is not a square. For prime fields, such an element is termed a quadratic nonresidue.
Case where the field has characteristic two
This case is a little trickier, since no polynomial of the form is irreducible. Rather, an irreducible polynomial is of the form .