Classifying extensions of the field of formal Laurent series over $\mathbb{F}_p$

Brown, Jim; Hasmani, Alfeen; Hiltner, Lindsey; Kraft, Angela; Scofield, Daniel; Wash, Kirsti

doi:10.1216/rmj-2015-45-1-115

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Absolute Values and Completions1

Finite Extension Of ( ) P T 1

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2023

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“…In this chapter, we consider two kinds of base field,  and ( ) q t  and introduce the notion of an absolute value on these base field and use this concept to construct an extension field. Most results, definition and theorems are taken from [12] [13] [14] [15].…”

Section: Absolute Values and Completionsmentioning

confidence: 99%

“…The t-adic absolute value can also be defined in terms of other parameters than p. Theorem 4.38. [12] The field ( )…”

Section: Finite Extension Of ( ) P T mentioning

confidence: 99%

“…21 [12]. For a prime p, define an absolute value on the rational numbers follows, for all x ∈  , let , ,m α β ∈  with is the p-adic absolute value on the rational numbers  .…”

mentioning

confidence: 99%

See 2 more Smart Citations

An Introduction to the Theory of Field Extensions

Chibeti¹,

Kyapwanyama²,

Phiri³

et al. 2023

APM

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This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to algebraic extensions. Finally, we construct finite extensions of  and finite extensions of the function field over finite field p  using the notion of field completion, analogous to field extensions. With the study of field extensions, considering any polynomial with coefficients in the field, we can find the roots of the polynomial, and with the notion of algebraically closed fields, we have one field, F, where we can find the roots of any polynomial with coefficients in F.

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Section: Absolute Values and Completionsmentioning

confidence: 99%

“…The t-adic absolute value can also be defined in terms of other parameters than p. Theorem 4.38. [12] The field ( )…”

Section: Finite Extension Of ( ) P T mentioning

confidence: 99%