Question

4.5 Define Try/F, (a) a +a ++a for any a E F. The element TrF/F, (a) is called the trace of a with respect to the extension F4/F. (i) Show that Trm/F, (a) is an element of F4 for all a e F (ii) Show that the map TrF F, (a) Tr/F, F F4 is an F-linear transformation, where both Fm and F are viewed as Fg vector spaces over (iii) Show that Tr/F, is surjective (iv) Let BE F. Prove that Tr/F, (B) = 0 if and only if there exists an element y e F such that By -y. (Note: this statement is commonly referred to as the additive form of Hilbert's Theorem 90.) (v) (Transitivity of trace.) Prove that TrFm/F, (a) TrF (Trm /F(a) for any α ε E".

Answer 1

(ii) For
a, BEF
and
a, bE
.

(aa b3) TrFm /E (aa b)(aa b6) (aa bB)

(aab)(aa) (bs) (aaT (b8)

(because we are working in a field of characteristic )

(aa)(aa) (aa) (b)(b3).(b)

Now since
a
, $a^{q^{k}}=a$ for
1k<m- 1
. and similar is true for .

Therefore, $\text{Tr}_{\mathbb{F}_{q^m}/\mathbb{F}_{q}}(a\alpha+b\beta)= a\alpha+a\alpha^{q}+...+a\alpha^{q^{m-1}}+b\beta+b\beta^{q}+...+b\beta^{q^{m-1}}$

So, $\text{Tr}_{\mathbb{F}_{q^m}/\mathbb{F}_{q}}(a\alpha+b\beta)=a\text{Tr}_{\mathbb{F}_{q^m}/\mathbb{F}_{q}}(\alpha)+ b\text{Tr}_{\mathbb{F}_{q^m}/\mathbb{F}_{q}}(\beta)$ .

Thus, the given transformation is linear.