Galois Theory

From MGSA
Jump to: navigation, search
  • Give a polynomial not solvable by radicals.
  • If α and β are algebraic over {\mathbb Q}, show that α + β is also algebraic over {\mathbb Q}.
  • If α is algebraic over {\mathbb Q}, show that {\mathbb Q}[\alpha] is a field.
  • Given a non-normal separable extension [E:F] = 4, bound the degree [K:F] of the normal closure of E. (Bergman)
  • What if there exists an intermediate field E\subseteq L \subseteq F of degree 2? Then what can ou say about Gal(F / E)? (Bergman)
  • Are there any Galois extensions whose group is \{\pm 1, \pm i, \pm j, \pm k\}? (Bergman)
  • What is an algebraic closure? (Lenstra)
  • What is an algebraically closed field? What does it mean when a polynomial splist? (Lenstra)
  • Show that "every polynomial over a field has a root in the field" implies that the field is algebraically close. (Lenstra)
  • What is the index of a field in its algebraic closure? (Lenstra)
  • What is [\overline{{\mathbb Q}}\colon \mathbb Q]? (Lenstra)
  • Give an example of an inseparable field extension.
  • What is a splitting field of a polynomial?
  • Let F be a field, and p(x)\in F[x] irreducible over F. Suppose that p(x) has roots α and β isn some extension field, with \alpha\not=\beta. Is it true that F(\alpha)\cong F(\beta)?
  • Is it true that a polynomial of degree n over a division ring has at most n roots in any extending division ring?
  • Let F be a field, θ an element (possibly transcendental) in an extension field E. The ring F[\theta]\cong F[x]/A for some ideal A. What can be said about A?
  • Give an example of a field F and a polynomial p(x) such that the splitting field E of p over F has no normal subextensions.
  • Show that a polynomial of odd degree always has at least one real root.
  • Let p(x) be a polynomial over {\mathbb Q} with Galois group \Z_4\times \Z_4. What can be said about the solvability of p(x) by radicals?
  • In a finite extension of fields, when can you conclude that the separable degree is equal to the degree?
  • How are normal extensions and splitting fields related?
  • Are all algebraic extensions finite?
  • Can the cardinality of an infinite field be increased by algebraic closure?
  • What is the Galois group of x8 − 1 over {\mathbb Q}?
  • What is the Galois group of x8 + 1 over {\mathbb Q}?
  • Define the concept of prime field.
  • Show that any two finite fields of the same order are isomorphic.
  • Let F be a Galois extension of degree n over k. What is the order of the Galois group of F over k?
  • Show that if F(a) is a finite extension of the field F, then the extension if algebraic.
  • Show that if F(a) is algebraic over F then [F(a):F] is finite.
  • Are extensions of degree 2 over {\mathbb Q} always normal?
  • Let t be transcendental over k, and let x={t^3+2\over t^2+3}. Is x algebraic over k?
  • What is a cyclotomic extension?
  • Construct a field of 27 elements. (Vojta)
  • Can you construct a field of 27 elements by starting with {\mathbb F}_3 and picking an irreducible polynomial of some degree? Which degree? (Vojta)
  • Prove that any two fields of 27 elements are isomorphic. (Vojta/Bergman)

This page was originally derived from this TeX file.

Retrieved from "http://www.ocf.berkeley.edu/~mgsa/wiki/Galois_Theory"
Personal tools