Field extensions are the algebraic setting for adjoining roots and measuring how much larger a field becomes. Degrees, minimal polynomials, and tower laws are the core computational tools.
Extensions and degrees
Definition. A field extension \(L/K\) means \(K\subset L\) and the operations of \(K\) agree with those inherited from \(L\). The degree \([L:K]\) is the dimension of \(L\) as a vector space over \(K\).
Example. \(\mathbb C/\mathbb R\) has degree \(2\), with basis \(1,i\). The extension \(\mathbb Q(\sqrt2)/\mathbb Q\) also has degree \(2\), with basis \(1,\sqrt2\).
Algebraic and transcendental elements
Definition. An element \(\alpha\in L\) is algebraic over \(K\) if it is a root of a nonzero polynomial in \(K[x]\). Otherwise it is transcendental over \(K\). The extension \(L/K\) is algebraic if every element of \(L\) is algebraic over \(K\).
Proposition. If \([L:K]<\infty\), then \(L/K\) is algebraic.
Proof. For \(\alpha\in L\), the vectors \(1,\alpha,\ldots,\alpha^n\) are linearly dependent over \(K\) once \(n\ge[L:K]\). A nontrivial linear relation gives a polynomial over \(K\) vanishing at \(\alpha\).
Minimal polynomials and simple extensions
Definition. If \(\alpha\) is algebraic over \(K\), its minimal polynomial \(m_{\alpha,K}\) is the unique monic irreducible polynomial in \(K[x]\) with \(m_{\alpha,K}(\alpha)=0\).
Theorem. If \(\alpha\) is algebraic over \(K\), then \[ K(\alpha)\cong K[x]/(m_{\alpha,K}), \qquad [K(\alpha):K]=\deg m_{\alpha,K}. \]
Proof. Evaluation \(K[x]\to K(\alpha)\), \(f\mapsto f(\alpha)\), has kernel \((m_{\alpha,K})\) and image \(K[\alpha]=K(\alpha)\) because \(\alpha^{-1}\), when \(\alpha\ne0\), is a polynomial in \(\alpha\) modulo the minimal polynomial.
Tower law
Theorem. If \(K\subset E\subset L\), then \[ [L:K]=[L:E][E:K] \] whenever the degrees are finite.
Proof. If \((u_i)\) is a basis of \(E/K\) and \((v_j)\) is a basis of \(L/E\), then the products \(v_ju_i\) form a basis of \(L/K\). Spanning and linear independence follow by grouping coefficients in \(E\).
A two-radical calculation
Calculation. The extension \(\mathbb Q(\sqrt2,\sqrt3)/\mathbb Q\) has degree \(4\). Since \(\sqrt3\notin\mathbb Q(\sqrt2)\), the tower law gives \[ [\mathbb Q(\sqrt2,\sqrt3):\mathbb Q]=2\cdot2=4. \] A basis is \(1,\sqrt2,\sqrt3,\sqrt6\).
Finite generation and algebraicity
Proposition. If \(\alpha_1,\ldots,\alpha_n\) are algebraic over \(K\), then \(K(\alpha_1,\ldots,\alpha_n)/K\) is finite and algebraic.
Remark. This proposition is one reason algebraic extensions behave like finite-dimensional linear algebra locally: every finitely generated subextension is finite.