Description:
Standard algebraic definitions for forms Form 241 through
Form 244
Content:
This note contains (standard) algebraic definitions for forms Form 241 through Form 244
Definition:
- If \(F\) is a subfield of \(E\), \(\alpha\in E\) is
algebraic over \(F\) if \(\exists a_0, a_1,\ldots, a_n\in F\) not all
zero such that \(a_0 + a_1\alpha \ldots + a_n\alpha^n = 0\).
- \(E\) is algebraic over \(F\) if every element of \(E\) is
algebraic over \(F\).
- A field \(L\) is algebraically closed if every polynomial
of degree \(\ge 1\) with coefficients in \(L\) has a root in \(L\).
- If \(K\) is a field, an algebraic closure of \(K\) is an
algebraically closed algebraic extension of \(K\).
- A field \(K\) is real if \(-1\) is not the sum of squares in
\(K\).
- A field is real closed if it is real and any real,
algebraic extension of \(K\) equals \(K\).
- A principal ideal domain is an integral domain
(commutative ring with identity and no zero divisors) in which every
ideal is principal.
- A unique factorization domain is an integral domain in
which
- every non-zero, non-unit element (unit \(\equiv \) invertible)
can be written \(a = c_{1}c_{2}\ldots c_{n}\) with \(c_{1}, c_{2}, \ldots ,
c_{n}\) irreducible. (That is, each \(c_{i}\) is non-zero and non-unit and
if \(c_{i} = ab\) then either \(a\) or \(b\) is a unit.)
- If \(a = c_{1}c_{2}\ldots c_{n} = d_{1}d_{2}\ldots
d_{m}\) where \(c_{i}\) and \(d_{i}\) are irreducible then \(n = m\) and
there is a permutation \(\sigma \) of \(\{1,2, \ldots ,n\}\) such that
\(c_{i} = r_{i}d_{\sigma (i)}\) where \(r_{i}\) is a
unit, \(i = 1, \ldots , n\)
Howard-Rubin number:
84
Type:
Definitions
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