💡 — Looking Ahead

In Part II of this course, Artin will restrict to commutative fields,
as Galois theory is traditionally developed in that setting. However, the linear
algebra of Part I is developed in full generality, allowing non-commutative fields,
which makes the results more powerful and broadly applicable.

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✏️ Example: Multi-Radical Extension

Let F = ℂ(x,y,z) and
K = F(⁴√(xyz), ⁴√(y²z), ⁴√(xz²)).
Then K/F is a 4-Kummer extension. The
subgroup of F^*/F✱⁴ generated by the cosets
of xyz, y²z, xz² has order 32 (verified by unique
factorization arguments comparing powers of x, y, z).
Therefore [K:F] = 32.

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📖 **Separable Extensions and Perfect Fields**

More generally:
A polynomial is
separable over if all its
irreducible factors are separable.
An algebraic element is
separable over if its
minimum polynomial is separable.
An algebraic extension is
separable if every
is separable over .
A field is perfect if every
polynomial in is separable over
.

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📐 **Theorem 3.7.10 (Kummer Theory)**

Let **F** ⊇ **F**₀(ζₙ) and
let **E** be an extension of
**F**. Then **E** is a
Kummer field if and only if:
**E** is a Galois extension of **F**.
Gal(**E**/**F**) is abelian.
The exponent of Gal(**E**/**F**) divides n.

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✏️ Applying the Complex Spectral Theorem

Consider the operator on ℂ² with matrix
String.rawA = (1 i \ -i 1)
Check: A^* = Āᵀ = A, so A is
self-adjoint (hence normal). The eigenvalues are
λ₁ = 0 and
λ₂ = 2, with orthonormal eigenvectors
String.rawe₁ = 1/√(2)(1 \ i), e₂ = 1/√(2)(1 \ -i)
The Spectral Theorem guarantees such an eigenbasis must exist for any normal operator.

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📖 **r Tag Markers**

write
User publishes their events to this relay. Clients downloading events
FROM a user should use their write relays.
read
User expects to receive mentions here. Clients sending events ABOUT
a user (where they're tagged) should use their read relays.
(no marker)
Relay is used for both reading and writing.

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🧮 **The Hermeneutical Invasion of Philosophy and Economics**

In recent decades, a new methodological approach has gained currency in the social sciences:
hermeneutics. Originally developed for the interpretation of texts, especially biblical and
legal texts, hermeneutics has been extended to the interpretation of all human phenomena.
Its influence has spread to economics, with troubling results.

— Rothbard — Economic Controversies

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✏️ The Dream of the Golden Country

"With what seemed a single movement she tore off her clothes and flung them
disdainfully aside... What overwhelmed him in that instant was admiration for
the gesture with which she had thrown her clothes aside. With its grace and
carelessness it seemed to annihilate a whole culture, a whole system of thought,
as though Big Brother and the Party and the Thought Police could all be swept
into nothingness by a single splendid movement of the arm."

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💡

Definition (Reducing Pair): Let τ ∈ L(V).
If V = S ⊕ T and if both S and T are
τ-invariant, we say that the pair (S, T) reduces τ.

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📖 **Subgroup**

A subset H of a group G is a subgroup if H is itself
a group under the operation inherited from G. We write H ≤ G.

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📐 **Corollary 5.2.2 (Extension of Isomorphisms)**

Let E be a normal extension of
F and let
'σ: B → B\' be any isomorphism of
intermediate fields with σ|_F = id.
Then σ extends to an automorphism
σ̃: E → E.

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💡 — The NIST P-256 Controversy

NIST's P-256 curve uses seemingly random constants generated by the NSA. Some
cryptographers worried these could hide backdoors. secp256k1's simple, deterministic
parameters inspired more trust in the cypherpunk community.

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