F(SuT)=F1(T) , where F1=F(S)
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Let F is subset of K be a field extension Let S and T be subsets of K Then
F(SuT)=F1(T) , where F1=F(S)
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- If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type here][Type here]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]Let where is a field and let . Prove that if is irreducible over , then is irreducible over .
- 4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .[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]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.