Mathematics Homework Solutions
Problem
#58242

Galois Theory : Show that any algebraic extension of a perfect field is perfect.

Show that any algebraic extension of a perfect field is perfect (using the below hint only).

HINT: Let K be a perfect field and F an algebraic extension of K. If F is not perfect, then there is a polynomial f(x) an element of F[x] that has an irreducible factor p(x) with a repeated root u. Here u is algebraic over K; let g(x) be the minimal polynomial of u over K. What is the relationship between g(x) and p(x)? Show that g(x) has a repeated root - this contradicts the hypothesis that K is perfect.


Solution Summary

Algebraic extensions of perfect fields are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Solution
What is this?
By OTA - Overall OTA Rating
Yupei Xiong, PhD - 4.8/5
Purchase Cost Now
$2.19 CAD (was ~$3.99)
Included in Download
  • Plain text response
$2.19 Instant Download
Add to Cart
Why you can trust BrainMass.com
  • Your Information is Secure
  • Best Online Academic Help Service
  • Students find real academic Success
Related Solutions
  • Perfect Numbers - An integer n is called k-perfect if σ(n) = kn (note that a perfect number is 2-perfect). (a) Show that 120 = 23• • • 3 • • • 5 is 3-perfect. (b) Show that if n is 3-perfect and gcd(3, n) = 1, ...
  • Show that every tree has at most one perfect matching. - Show that every tree has at most one perfect matching. Can you explain what does perfect matching mean and draw a graph please.
  • Primes and divisibility word problem - When the accountants for lose-a-digit Computer, Inc. had finished preparing their annual budget, they presented the final figures to the president, I.M. Smart. "It looks like a good year," he exclaime ...
  • Compact and perfect sets - If P is a perfect set and K is compact is the intersection P intersection K always compact?always perfect?.
  • Perfect square - Find a value for k that will make 4x^2 + 6.4 x + k a perfect square. Show the procedure that you used which requires algebra [that is, not trial and error].
Browse