Directions. It is intended that you work these as exercises. Although you may refer to books for definitions and standard theorems, searching for solutions to these written exercises either in books or in online references should not be required and is undesirable. If you make use of a reference other than class notes, you must properly cite that use. You may not seek help from others.
Let be the splitting field over of the polynomial , and let .
Show that is the splitting field of over .
Find the Galois group of over .
Find the Galois group of over .
Find the Galois group of over .
Let be a field, a (finite) Galois extension of , the Galois group, and the group ring of with coefficients in . Observe that and have the same dimension as vector spaces over .
Show that there is an obvious -module structure on .
Formulate in terms of the study of as an extension of without reference to the notion of group ring what it means for to be isomorphic as an -module to .
How small can a non-commutative ring be? Prove the following:
Every ring with is commutative.
If is the field of elements, there is a subring of the matrix ring over that is non-commutative and has elements.
In Hungerford's text at p. 227, Defn. 7.1, an “algebra” over a commutative ring is defined to be a ring , not necessarily commutative, that is also a -module where it is required that the -module structure is related to the ring multiplication of by the formulae
Prove that there is a ring homomorphism such that for all and one has (Be sure to begin by defining .)
Let be any field, and let be the field of rational functions in variables over . Find a subfield of with the property that is a cyclic extension of of degree .