𝔽-valued trace of a finite-dimensional commutative 𝔽-algebra
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A non-zero 𝔽-valued 𝔽-linear map on a finite dimensional 𝔽-algebra is called an 𝔽-valued trace if its kernel does not contain any non-zero ideals. However, given an 𝔽-algebra such a map may not always exist. We find an infinite class of finite-dimensional commutative 𝔽-algebras which admit an 𝔽-valued trace. In fact, in these cases, we explicitly construct a trace map. The existence of an 𝔽-valued trace on a finite dimensional commutative 𝔽-algebra induces a non-degenerate bilinear form on the 𝔽-algebra which may be helpful both theoretically and computationally. In this article, we suggest a couple of applications of an 𝔽-valued trace map of an 𝔽-algebra to algebraic coding theory.
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