Let F be a finite field of order q and let n ∈ Z+. Prove that |GLn(F ) : SLn(F )|= q − 1.
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Let F be a finite field of order q and let n ∈ Z+. Prove that |GLn(F ) : SLn(F )|= q − 1.
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- Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].Let F be a field and f(x)=a0+a1x+...+anxnF[x]. Prove that x1 is a factor of f(x) if and only if a0+a1+...+an=0. Prove that x+1 is a factor of f(x) if and only if a0+a1+...+(1)nan=0.[Type here] True or False Label each of the following statements as either true or false. 3. Every integral domain is a field. [Type here]
- Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.Suppose S is a subset of an field F that contains at least two elements and satisfies both of the following conditions: xS and yS imply xyS, and xS and y0S imply xy1S. Prove that S is a field. This S is called a subfield of F. [Type here][Type here]Label each of the following as either true or false. If a set S is not an integral domain, then S is not a field. [Type here][Type here]