Algebraic Element
المؤلف:
المرجع الالكتروني للمعلوماتيه
المصدر:
المرجع الالكتروني للمعلوماتيه
الجزء والصفحة:
...
17-1-2019
1344
Algebraic Element
Given a field 
and an extension field 
, an element 
is called algebraic over 
if it is a root of some nonzero polynomial with coefficients in 
.
Obviously, every element of 
is algebraic over 
. Moreover, the sum, difference, product, and quotient of algebraic elements are again algebraic. It follows that the simple extension field 
is an algebraic extension of 
iff 
is algebraic over 
.
The imaginary unit i is algebraic over the field 
of real numbers since it is a root of the polynomial 
. Because its coefficients are integers, it is even true that 
is algebraic over the field 
of rational numbers, i.e., it is an algebraic number (and also an algebraic integer). As a consequence, 
and 
are algebraic extensions of 
and 
respectively. (Here, 
is the complex field 
, whereas 
is the total ring of fractions of the ring of Gaussian integers ![Z[i]](http://mathworld.wolfram.com/images/equations/AlgebraicElement/Inline23.gif)
.)
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