# algebraic extension of a field

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## algebraic extension of a field

[¦al·jə¦brā·ik ik′sten·shən əv ə ′fēld]
(mathematics)
A field which contains both the given field and all roots of polynomials with coefficients in the given field.
References in periodicals archive ?
Here we are interested in certain local global principles related with the splitting property of a given reductive group defined over an infinite algebraic extension of a global field.
Let L be a field, k an algebraic extension of L contained in an algebraic closure of L with the set of all places [V.sub.k].
Then the algebraic extension K([X.sup.1],...,[X.sub.n]) / [K.sup.e]([X.sub.1],...,[X.sub.n]) has a basis composed only of units in [O.sub.w].
It is a well known fact that the algebraic extension K([X.sub.1],...,[X.sub.n]) / [K.sup.e]([X.sub.1],...,[X.sub.n]) has degree n!.
A valued field (Kv) is called henselian if the valuation v can be uniquely extended to each algebraic extension of the field K.
Observe that even for R = K(i) being the algebraic extension of a field K (not containing i) by i the coefficients of polynomials P and Q belong to R and not necessarily to K, and so the matrices above need not to be hermitian.
If the complete linearly ordered ternar in this theorem is obtained by the above extension of the ternar of a given betweenness plane, then the new betweenness plane, constructed according to this theorem, can be seen as an algebraic extension of the given plane.
If considering the projective plane obtained by algebraic extension, one has here a collineation in the sense of the theory of projective geometry.
Let k be an infinite global field, which is an infinite algebraic extension of a global field F.
completion) of an infinite algebraic extension of a global field k and G a semisimple simply connected algebraic group defined over k(v).
(A Risch algorithm uses induction on the number of monomial and algebraic extensions required to construct the function field containing the integrand beginning with the rational functions as base field, .
of Science) explore anti-integral, super-primative and ultra-primitive extensions along with other subtopics such as flatness, integrality and "unramifiedness." They focus on simple algebraic extensions and show that simple extensions of a Noetherian domain R can be complicated even if they are bi-rationally equal to R.
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