🏗️ ΘρϵηΠατπ🚧 (under construction)

Automorphism
If E is a field, then an automorphism of E is an isomorphism of E with itself.
Automorphism Fixed Pointwise
If E/F is a field extension, then an automorphism σ of E fixes F pointwise if σ(c)=c for every cF
Automorphism Fixed Pointwise Takes Roots to Roots
Let f(x)F[X] and let E/F be an extension field of F. If σ:EE is an automorphism fixing F pointwise , and if αE is a root of f(x) then σ(α) is also a root of f(x)
Galois Group
Let E/F be a field extension, then it's Galois Group is Gal(E/F):={ automorphisms σ of E fixing F pointwise } where the
Degree 1 iff Same as Fields
Suppose that E is a field extension of F, then [E:F]=1E=F
Prime Degree Extension Restricts Subfields
Suppose that the degree of the extension E/F is a prime p. Show that any subfields K of E containing F is either E or F.
Irreducible Polynomial Restricts Size of Galois Group
Suppose that p(x) is irreducible fo degree n, and let E be p(x)'s splitting field then
  • n|Gal(E/Q)|n!
  • |Gal(E/Q)|n!
Polynomial to Galois
Consider the polynomial given by p(x)=x3+3x1Q[x]
  • Show that p(x) has exactly one real root given by αR
  • Show that Q(α) is a field extension of Q of degree 3
  • Let E be the splitting field of p(x) over Q find the size of the Galois group G(E/Q) of E/Q.
  • For each isomoprhism ϕGal(E/Q) determine ϕ(α)
The Irreducible Polynomial of 1 + i
Find the irreducbile polynomial of 1+i over Q
Degree of Extension
Find the degree of the extension Q(3+22)
Quartic Roots
Find the order of the Galois group Gal(Q(24,i)/Q(2)) and say what each of its elements are doing to the generators 24 and i.
Square Root Coefficients
Let f(x)=x32Q(2)[x]
  • Show that f(x) is irreducible over Q(2)
  • Let E be the splitting field of f(x). Find Gal(E/Q(2)) and describe its elements.
Polynomials as Matrices
Let p(x)=a0+a1x1++amxmF[x] be a polynomial with coefficients in a field F. Let AMn×n(F), and note that we can evaluate p(A)=a0I+a1A++amAm
  • Prove that there is a non-zero polynomial p(x)F[x] such that p(A)=0
  • Let I be the set of all polynomaisl q(x)F[x] such that q(A)=0, prove that I is an ideal in F[x]
  • Let m(x) be a monic polynomilal of least degree in I. Prove that I=(m(x)).
  • Prove that m(x) is the unique monic polynomial of least degree in I
Big Adjoin
Suppose E=(α1,α2,,αn) where αi2 for i=1,2,,n. Prove that 23E.
Automorphism Takes Roots of a Factor to the Root of the Same Factor
Let f(x)F[x] such that f(x)=a(x)b(x) where a(x),b(x) are irreducible and let E/F be a field extension of F. If σ:EE is an automorphism fixing F pointwise, and if αE is a root of f(x) such that a(α)=0 then σ(α) is a root of f(x) which makes a(σ(α))=0
Galois Group of Factored Polynomial
Let f(x)=(x22)(x23)(x25)Q[x] and E=Q(2,3,5) be the splitting field over f(x) over Q. Find the galois group Gal(E/Q) of f(x).
A Familiar Group
Consider the polynomial f(x)=x45Q[x] and let E=Q(i,54) be it's splitting field over Q
  • Show that |Gal(E/Q)|=[E:Q]=8
  • Describe all automorphisms {σ1,σ2,σ8} in Gal(E/Q)
  • Write the multiplication table for the group Gal(E/Q)
  • Which finite group is Gal(E/Q) isomorphic to?
Only One
Let B=(i54). Recall (see Q.3) that B is a subfield of the splitting field E of f(x)=x45[x].
  • Show that |Gal(E/B)|=2.
  • Describe the nontrivial automorphism σGal(E/B). That is say what σ is doing to the generators of E.
Big Polynomial
Let f(x)=x66x410x3+12x260x+17Q[x] and let E=Q(α1,α2,α3,α4,α5,α6) where
  • α1=2+53
  • α2=2+ω53
  • α3=2+ω253
  • α4=2+53
  • α5=2+ω53
  • α6=2+ω253

Show that
  • E=Q(2,53,ω)
  • Show that |Gal(E/Q)|=[E:Q]=12
  • Describe all the automorphisms {σ1,σ2,...σ12} in Gal(E/Q)
Cos 20 Is not a Surd
In this question we only consider subfields of . Recall that a field extension B/F is a pure extension of type m=2 if B=F(α), where α2F but αF. A tower of number fields F=B0B1 Bt is a radical tower were each Bi+1/Bi is a pure extension of type 2 and F=. Let S denote the union of all towers of number fields. The numbers in S are called surds. So a surd is a real number that is in some tower of number fields.
  • Prove that S is a field. Note that S.
  • Let B/ be a field extension. Let p(x)B[x] be a polynomial with coefficients in B. Suppose a+brB(r) is a root of p(x), where rB but rB. Show that abr is also a root of p(x). Hint: apply the 'conjugation' σ to the equation p(a+br)=0. Here σ denotes the automorphsm of B(r) fixing B pointwise, determined by σ(a+br)=abr.
  • Let p(x) be a cubic polynomial with rational coefficients. Show that the sum of the roots of p(x) is a rational number.
  • Using the notation of part (b), assume p(x)B[x] is a cubic polynomial. Prove that if a+brB(r) is a root of p(x) then p(x) also has a root in B. Hint: parts (b), (c).
  • Use induction to prove that if a cubic polynomial p(x)[x] has a surd root, it must have a rational root.
  • Let θ denote a 20 degree angle. Show that (2cosθ)33(2cosθ) 1=0.
  • Show that x0:=2cosθ is not a surd. Hint: by part (f), we know x0 is a root of the cubic polynomial p(x)=x33x1[x]. So if x0 where a surd then p(x) would have a rational root. Show that p(x) has no rational roots (can use rational root theorem).