# Problems in Galois Theory

Not what you're looking for?

a. Let K be a field of characteristic p > 0, and let c in K. Show that if alpha is a root of f (x) = x^p - x - c, so is alpha + 1. Prove that K(alpha) is Galois over K with group either trivial or cyclic of order p.

b. Find all subfields of Q ( sqrt2, sqrt 3) with proof that you have them all. What is the minimal polynomial of sqrt2+ sqrt3? Which subfields does it generate over Q?

##### Purchase this Solution

##### Solution Summary

The expert solves two problems in Galois theory. Which subfield generates over Q is determined.

##### Solution Preview

Please see the attachment.

a. Let K be a field of characteristic p > 0 and let c E K. Show that if alpha is a root of f(x) = x^p - x -c, so is alpha + 1. Prove that K(alpha) is Galois over K with group either trivial or cyclic of order p.

We have:

f(alpha+1) = (alpha+1)^p - (alpha+1) - c

= alpha^p + 1 - alpha - 1 - c

= alpha^p - alpha - c = 0

Whence alpha + 1 is a root of f whenever alpha is a root of f. Thus we see that if f has at least one root, then it has p roots, i.e. it splits over K, in which case Gal(K(alpha)/K) is trivial. On the other hand, if f is irreducible over K, then ...

##### Purchase this Solution

##### Free BrainMass Quizzes

##### 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.

##### Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

##### Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

##### Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

##### Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.