For a Galois extension field of a field , the fundamental theorem of Galois theory states that the subgroups of the Galois group correspond with the subfields of containing . If the subfield corresponds to the subgroup , then the extension field degree of over is the group order of ,
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Suppose , then and correspond to subgroups and of such that is a subgroup of . Also, is a normal subgroup iff is a Galois extension field. Since any subfield of a separable extension, which the Galois extension field must be, is also separable, is Galois iff is a normal extension of . So normal extensions correspond to normal subgroups. When is normal, then
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as the quotient group of the group action of on .
According to the fundamental theorem, there is a one-one correspondence between subgroups of the Galois group and subfields of containing . For example, for the number field shown above, the only automorphisms of (keeping fixed) are the identity, , , and , so these form the Galois group (which is generated by and ). In particular, the generators and of are as follows: maps to , to , and fixes ; maps to , to and fixes ; and maps to , to and fixes .
For example, consider the Galois extension field
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over , which has extension field degree six. That is, it is a six-dimensional vector space over the rationals.