MHB Proving Irreducibility of x^4 −7 Using Polynomial Theorems

Poirot1 · 2012-03-11T19:53:16+00:00

[FONT=CMR10][FONT=CMR10]Explain why the polynomial x^4[FONT=CMR7][FONT=CMR7] [FONT=CMSY10][FONT=CMSY10]−[FONT=CMR10][FONT=CMR10]7 is irreducible over [FONT=MSBM10][FONT=MSBM10]Q[FONT=CMR10][FONT=CMR10], quoting any theorems you use.

Thread starter Poirot1
Start date Mar 11, 2012
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Polynomial

Mar 11, 2012

Poirot1

Explain why the polynomial x^4 −7 is irreducible over Q, quoting any theorems you use.

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Mar 11, 2012

tarantella

Which theorems related to irreducibility do you know?

Thread 'About the existence of Hamel basis for vector spaces'

It is well known that a vector space always admits an algebraic (Hamel) basis. This is a theorem that follows from Zorn's lemma based on the Axiom of Choice (AC). Now consider any specific instance of vector space. Since the AC axiom may or may not be included in the underlying set theory, might there be examples of vector spaces in which an Hamel basis actually doesn't exist ?

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