Extensions of A Field
An extension of a field k is just a field K containing k as a subfield. One distinguishes between extensions having various qualities. For example, an extension K of a field k is called algebraic, if every element of K is a root of some polynomial with coefficients in k. Otherwise, the extension is called transcendental.
The aim of Galois theory is the study of algebraic extensions of a field.
Read more about this topic: Field Theory (mathematics)
Famous quotes containing the words extensions of, extensions and/or field:
“The psychological umbilical cord is more difficult to cut than the real one. We experience our children as extensions of ourselves, and we feel as though their behavior is an expression of something within us...instead of an expression of something in them. We see in our children our own reflection, and when we don’t like what we see, we feel angry at the reflection.”
—Elaine Heffner (20th century)
“The psychological umbilical cord is more difficult to cut than the real one. We experience our children as extensions of ourselves, and we feel as though their behavior is an expression of something within us...instead of an expression of something in them. We see in our children our own reflection, and when we don’t like what we see, we feel angry at the reflection.”
—Elaine Heffner (20th century)
“The woman ... turned her melancholy tone into a scolding one. She was not very young, and the wrinkles in her face were filled with drops of water which had fallen from her eyes, which, with the yellowness of her complexion, made a figure not unlike a field in the decline of the year, when the harvest is gathered in and a smart shower of rain has filled the furrows with water. Her voice was so shrill that they all jumped into the coach as fast as they could and drove from the door.”
—Sarah Fielding (1710–1768)