Field Extensions: Degrees, Algebraicity, Galois Theory, and Constructions

All flashcards

What is the degree of a field extension $L/K$?

The dimension of $L$ as a vector space over $K$.

What is a simple extension?

An extension of the form $K(s)/K$, obtained by adjoining one element $s$.

What is the minimal polynomial of an algebraic element $x$ over $K$?

The monic polynomial of least degree in $K[X]$ having $x$ as a root.

When is an algebraic extension normal?

When every irreducible polynomial over the base field with a root in the extension splits there.

What is a splitting field of a polynomial over $K$?

An extension in which the polynomial factors into linear factors.

Quiz questions

1. Which conditions characterize a subfield $K$ of a field $L$?Answer: It contains $1$ and is closed under addition, subtraction, multiplication, and nonzero inverses.
2. How is the notation $L/K$ read in field theory?Answer: “$L$ over $K$,” indicating that $L$ is an extension field of $K$.
3. If $K\subseteq F\subseteq L$, what is $F$ called relative to $L/K$?Answer: An intermediate field.
4. Why does a subfield $K$ of $L$ contain the same zero element as $L$?Answer: It contains $1$ and is closed under subtraction, so it contains $1-1=0$.
5. Why can identifying a field with a subfield through an embedding require care?Answer: More than one injective field homomorphism may give different identifications.