By David S. Dummit, Richard M. Foote

"Widely acclaimed algebra textual content. This ebook is designed to offer the reader perception into the ability and wonder that accrues from a wealthy interaction among varied components of arithmetic. The publication conscientiously develops the idea of alternative algebraic buildings, starting from easy definitions to a few in-depth effects, utilizing various examples and workouts to help the reader's knowing. during this approach, readers achieve an appreciation for a way mathematical constructions and their interaction bring about strong effects and insights in a couple of diverse settings."

Covers primarily all undergraduate algebra. Searchable DJVU.

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**Example text**

On the other hand, if β ∈ B1 ∩ B2 , then b ∈ B1 , so σ (β) = β for every σ ∈ H1 , and b ∈ B2 , so σ (β) = β for every σ ∈ H2 ; hence σ (β) = β for every σ ∈ H3 . In the original development of group theory, and in particular in the work of Galois, groups were not regarded as abstract groups, but rather as groups of symmetries, and in particular as groups of permutations of the roots of a polynomial. This viewpoint, still a most useful one, is encapsulated in the following result. In order to state it, we first recall that an action of a group G on a set S = {si } is transitive if for any two elements si and s j of S, there is an element σ of G with σ (si ) = s j .

Suppose (D/F) is finite. Then (BD/B) ≤ m (D/F) and BD = { i=1 bi di | bi ∈ B, di ∈ D}. Furthermore, (BD/F) ≤ (B/F)(D/F). m Proof. Let (BD)0 = { i=1 bi di }. If {δi } is a basis for D as an F-vector space, then {δi } spans (BD)0 as a B-vector space, so dimB (BD)0 ≤ dimF D. 3, and hence BD = (BD)0 . 1, (BD/F) = (BD/B)(B/F) ≤ (D/F)(B/F), as claimed. One of the most common, and important, ways of obtaining field extensions is by “adjoining” elements. 5. Let E be an extension of F, and let {αi } be a set of elements of E.

2) Let E be an algebraic extension of F. Then α ∈ E is a separable element if m α (X ) is a separable polynomial. Otherwise, α is an inseparable element. (3) Let E be an algebraic extension of F. Then E is a separable extension of F if every α ∈ E is a separable element. Otherwise, it is an inseparable extension. 4. We are most interested in extensions that are both normal and separable, as these have the closest connection with Galois theory, so we will defer our consideration of extensions that satisfy only one of these two properties until later.