1
Compute each of the following Galois groups. Which of these field extensions are normal field extensions? If the extension is not normal, find a normal extension of in which the extension field is contained.
(a) (c)
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Compute each of the following Galois groups. Which of these field extensions are normal field extensions? If the extension is not normal, find a normal extension of in which the extension field is contained.
(a) (c)
Determine the separability of each of the following polynomials.
over
over
over
over
(a) Separable over since (c) not separable over since
Give the order and describe a generator of the Galois group of over
If
then A generator for is where for
Find a primitive element in the splitting field of each of the following polynomials in
(a)
Prove that the Galois group of an irreducible quadratic polynomial is isomorphic to
Prove that the Galois group of an irreducible cubic polynomial is isomorphic to or
Let be the splitting field of a cubic polynomial in Show that is less than or equal to 6 and is divisible by 3. Since is a subgroup of whose order is divisible by 3, conclude that this group must be isomorphic to or
Let be fields. If E is a normal extension of show that must also be a normal extension of
Let be the Galois group of a polynomial of degree Prove that divides
is a subgroup of
Let If is solvable over show that is also solvable over
Construct a polynomial in of degree 7 that is not solvable by radicals.
Let be prime. Prove that there exists a polynomial of degree with Galois group isomorphic to Conclude that for each prime with there exists a polynomial of degree that is not solvable by radicals.
Let be a prime and be the field of rational functions over Prove that is an irreducible polynomial in Show that is not separable.
Let be an extension field of Suppose that and are two intermediate fields. If there exists an element such that then and are said to be Prove that and are conjugate if and only if and are conjugate subgroups of
Let If is a positive real number, show that
Let be the splitting field of Prove or disprove that is an extension by radicals.
True.
Let be a field such that Prove that the splitting field of is where
Prove or disprove: Two different subgroups of a Galois group will have different fixed fields.
Let be the splitting field of a polynomial over If is a field extension of contained in and then is the splitting field of some polynomial in
We know that the cyclotomic polynomial
is irreducible over for every prime Let be a zero of and consider the field
Show that are distinct zeros of and conclude that they are all the zeros of
Show that is abelian of order
Show that the fixed field of is
Clearly are distinct since or 0. To show that is a zero of calculate
The conjugates of are Define a map by
where Prove that is an isomorphism of fields. Show that generates
Show that is a basis for over and consider which linear combinations of are left fixed by all elements of
Let be a finite field or a field of characteristic zero. Let be a finite normal extension of with Galois group Prove that if and only if
Let be a field of characteristic zero and let be a separable polynomial of degree If is the splitting field of let be the roots of in Let We define the of to be
If show that
If show that
Prove that is in
If is a transposition of two roots of show that
If is an even permutation of the roots of show that
Prove that is isomorphic to a subgroup of if and only if
Determine the Galois groups of and