Fields Extensions/Algebraic
Not what you're looking for?
Let F be an extension field of K. Clearly F is a vector space over K. Let u be an element of F. Show that the subspace spanned by {1, u, u^2, ...} is a field IF and ONLY IF (iff) u is algebraic over K.
Let S be the subspace of F.
Hint for the "-->" of proof. If S is a field and u is not equal to 0, then 1/u is in S. What does that imply?
Hint for the "<---" Show S is closed under +, -, and x. Then need to show that if v is an element of S with v not equal to 0, then 1/v is an element of S. Suppose u is algebraic over K with minimal polynomial p(x) then consider how multiplicative inverses are found in K[x]/<p(x)>.
Purchase this Solution
Solution Summary
This solution is comprised of a detailed explanation to show that the subspace spanned by {1, u, u^2, ...} is a field IF and ONLY IF (iff) u is algebraic over K.
Solution Preview
Mathematics, Algebra
Fields Extensions/Algebraic
Let F be an extension field of K. Clearly F is a vector space over K. Let u be an element of F. Show that the subspace spanned by { 1, u, u2, ...} is a field IF and ONLY IF (iff) u is algebraic over K.
Hint for the "-->" of proof. Let S be the subspace of F. If S is a field and u is not equal to 0, then 1/u is in S. What does that imply?
Hint for the "<---" Show S is closed under +, -, and x. Then need to show that if v is an element of S with v not equal to 0, then 1/v is an element of S. Suppose u is algebraic over K with minimial polynomial p(x) then consider how multiplicative inverses are found in K[x]/<p(x)>. ...
Purchase this Solution
Free BrainMass Quizzes
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.