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------------------------------------------------ Let [tex]f[/tex] be a field with 76 elements, and let [tex]k[/tex] be a subfield of [tex]f[/tex] with 49 elements. What is the dimension of [tex]f[/tex] as a vector space over [tex]k[/tex]?

Answer :

Final answer:

The dimension of f as a vector space over k is 6.

Explanation:

The dimension of f as a vector space over k can be determined by finding the ratio of the cardinalities of f and k as fields. Since f has 76 elements and k has 49 elements, the dimension of f over k is calculated as:

dimkf = log7(76)

= log7(72)3

= log7(7)6

= 6

Therefore, the dimension of f as a vector space over k is 6.

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