MATH 431 activity: Galois groups and fixed fields
Let K be the splitting field of some polynomial f (x) ∈ F [x] with no multiple roots in K. In this activity
we want to understand the connections between the subgroups of Gal(K/F ) and the fixed fields of those
subgroups. With a partner or two, answer the following questions.
√
√
K = Q( −5). For questions (1)-(6) consider K = Q( −5) as an extension of Q.
(1) Fill in the subfield diagram for K below. Mark each extension with its degree.
Q
(2) Find all elements of Gal(K/Q). (Define each automorphism by where it sends
√
−5.)
(3) Which familiar group is Gal(K/Q) isomorphic to?
(4) Construct the subgroup diagram for Gal(K/Q).
(5) For each subgroup on your diagram (including the whole group and the trivial group), find the fixed field
of that subgroup (i.e., find KH for all H a subgroup of Gal(K/Q)). Make a list of these fixed fields.
(6) What do you notice about the structure of the subfield diagram and the subgroup diagram?
√ √
√ √
L = Q( 2, 3). For questions (7)-(11) consider L = Q( 2, 3) as an extension of Q.
(7) Fill in the subfield diagram for L below. Mark each extension with its degree.
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Q( 6)
Q
(8) Find all elements of Gal(L/Q). (Define each automorphism by where it sends
√
2 and
√
3.) What familiar
group Gal(L/Q) is isomorphic to? (You could consider element orders or construct the Cayley table for
the group.)
(9) Construct the subgroup diagram for Gal(L/Q). (Name the subgroups whatever you want.)
(10) For each subgroup on your diagram (including the whole group and the trivial group), find the fixed field
of that subgroup (i.e., find LH for all H a subgroup of Gal(L/Q)). Make a list of these fixed fields.
(11) What do you notice about the structure of the subfield diagram and the subgroup diagram?
(12) Now compare the subfield diagrams with the subgroup diagrams in questions 1 and 4, and in questions 7
and 10. Describe the relative position of any subgroup H of the Galois group in the subgroup diagram and
its fixed field in the subfield diagram.
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(13) Do the process demonstrated for K and L again for M = Q( 3 2, ω3 ), where ω3 is a primitive third root of
unity. (Find the subfield diagram, the Galois group, the subgroup diagram, and the fixed fields.)
Questions
Let E be a splitting field over F . Consider a subfield K of E and a subgroup H of Gal(E/F ).
(1) |Gal(E/F )| =?
(2) If EH = K, |H| =?
(3) In each of our examples, count the number of subfields strictly between Q and the full extension.
Then count the number of subgroups strictly between the full Galois group and the trivial subgroup
{id}. How are these related?
(4) True or False? Every subfield of E is the fixed field of some element of Gal(E/F ).
(5) True or False? Every subfield of E is the fixed field of some subgroup of Gal(E/F ).
(6) True or False? Two distinct elements of Gal(E/F ) can have the same fixed field.
(7) True or False? Two distinct subgroups of Gal(E/F ) can have the same fixed field.
Based on questions (3)-(7), there exists a
the subfields of E containing F and the subgroups of Gal(E/F ).
(8) If EH = K, H = Gal(
(9) [E : K] = |?|
(10) [K : F ] =
|?|
|?|
/
).
(phrase) between