Sentence examples for maximal subfield of from inspiring English sources

Conversely, any maximal subfield of (Delta ^{prime }) is (k -isomorphic to a maximal subfield of (Delta ).

Let (P) be a maximal subfield of (Delta ).

We have begin{aligned} d geqslant [F : K] geqslant [P : k] = d, end{aligned}which implies that (F) is a maximal subfield of (mathcal D ).

A subsequent paper by Noether showed, as a special case of a more general theorem, that all maximal subfields of a division algebra D are splitting fields.

Define(mathbf{gen}^{prime }(D, Gamma ))to be the collection of classes([D^{prime }] in mathrm{Br}(K where(D^{prime })is a central quaternion division(K -algebra with the following property: any maximal subfield(P)of(D)that is generated by an element of(Gamma )admits a(K -algebrang into(D^{prime }).

It follows that (Delta _1) contains a maximal subfield (mathcal P ) which is ramified over (mathcal L ^{(1)}_2).

One can generalize the notion of genus from division algebras to algebraic groups using maximal tori in place of maximal subfields.

If (k) satisfies the following property: If (D) and (D^{prime }) are central division (k -algebras of exponent 2 having the same maximal subfields then (D simeq D^{prime })in other words, for any (D) of exponent 2, (vert mathbf{gen}(D) cap _2mathrm{Br}(k -algebras)).

Then the field of rational functions (k(x)) also satisfies If (D) and (D^{prime }) are central division (k -algebras of exponent 2 having the same maximal subfields, then (D simeq D^{prime }) (in other words, for any D of exponent 2, (vert mathbf{gen}(D) cap _{2}{mathrm{Br}}(k -algebras)).

(1) Suppose (k) satisfies the following property: If (D) and (D^{prime }) are central division (k -algebras of exponent 2 having the same maximal subfields, then (D simeq D^{prime }) (in other words, for any D of exponent 2, (vert mathbf{gen}(D) cap _{2}{mathrm{Br}}(k -algebras)).

(1) If (k) satisfies the following property: If (D) and (D^{prime }) are central division (k -algebras of exponent 2 having the same maximal subfields then (D simeq D^{prime })in other words, for any (D) of exponent 2, (vert mathbf{gen}(D) cap _2mathrm{Br}(k -algebras)).