ΜΕΜ-227 Field Theory (Εαρινό 2026)

Section outline

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  Course Information

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  Instructor

  Theodoulos Garefalakis

  Office	Γ216
  email	tgaref@uoc.gr

   

  Weekly Schedule

  Lecture	Monday and Friday 11:00-13:00 (Room E204)

   

  Reading Material

  1. J. Rotman, Galois Theory, 2nd ed., Springer
  2. I. Stewart, Galois Theory, 5th ed., CRC Press.
  3. P. Morandi, Field and Galois Theory, Springer.
  4. D.S. Dummit and R.M. Foote, Abstract Algebra, Wiley.

   

  Evaluation

  There will be a midterm exam and a final exam. The final mark is computed as

  $\text{Mark} = \max\{ 0.3\cdot \text{Midterm} + 0.7\cdot \text{Exam}, \text{Exam}\}$

  The mark of the midterm does not count towards other evaluation periods (e.g. September, January examinations).

  Midterm: The midterm exam is scheduled for Monday 23/3/2026, 11:00 - 13:00 (during class).

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    Notes File
  • Select activity Sample Midterm Exam

    

    Sample Midterm Exam File
• Select section 9 February - 15 February

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  9 February - 15 February

  We reviewed some basic structures and theorems that will be needed throughout the course. In particular, we recalled the notions of ring, domain, field, ideal of a domain, principal ideal domain (PID), prime element and prime ideal, irreducible element and maximal ideal.

  We proved Eisenstein's irreducibility criterion and saw Gauss' Lemma on the irreducibility of polynomials over unique factorization domains and their field of fractions.

  We saw that a field K is a vector space over any subfield F, and defined degree of the extension K/F to be the dimension of K as an F-space. We denote it by $[K : F]$. We saw examples.

  If $K/F$ is a field extension and $X \subseteq K$, we defined the extension of $F$ that is generated by $X$ over $F$, and denoted is by $F(X)$. We saw $F(X)$ for finite sets $X$.

  We defined the notion of a algebraic element $\alpha$ over $F$, and of an algebraic extension $K/F$. For $\alpha$ algebraic over $F$, we defined the minimal polynomial $\min(F, \alpha)$ and proved some basic facts.

  We studied the simple extension $F(\alpha)/F$ for $\alpha$ algebraic over $F$.

  Reading: pages 1 - 5 from the notes.

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    Problem Set 1 File
• Select section 16 February - 22 February

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  16 February - 22 February

  We continued our study of algebraic extensions.

  We proved that $K/F$ is finite if and only if it is algebraic and finitely generated.

  For a tower $F \subseteq L \subseteq K$, we proved that $K/F$ is algebraic if and only if $K/L$ and $L/F$ are algebraic. In case they are finite, we proved that $[K:F] = [K:L] \cdot [L:F]$.

  We defined the splitting field of a polynomial $f \in F[x]$ over the field $F$. We proved its existence. We saw examples.

  Reading: pages 5-7 from the notes. 

  Reading: pages 7-9 from the notes.

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    Problem Set 2 File
• Select section 23 February - 1 March

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  23 February - 1 March

  Monday was a holiday.

  We discussed problem sets 1 and 2.
• Select section 2 March - 8 March

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  2 March - 8 March

  We defined the notion of an algebraically closed field and the algebraic closure of a field. 

  We discussed the ruler and compass constructions and their limitations.

  We started our study of embeddings $\tau : K \rightarrow L$. We saw how an isomorphism $\sigma : F \rightarrow F'$ can be extended to an isomorphism $\tau : F(\alpha) \rightarrow F'(\alpha')$. 

  Reading: pages 10-11 from the notes

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    Problem Set 3 File
• Select section 9 March - 15 March

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  9 March - 15 March

  We proved that any two splitting fields, $K, K'$, of a polynomial in $f \in F[x]$, over $F$ are isomorphic. More generally, if $\sigma : F \rightarrow F'$ is an isomorphism, $f \in F[x]$, and $K, K'$ are splitting fields of $f$ and $\sigma^*(f)$ over $F, F'$ respectively, then $\sigma$ can be extended to $\tau : K \rightarrow K'$. If $\alpha \in K$ is a root of $f$, and $\alpha' \in K'$ is a root of $\sigma^*(\min(F, \alpha))$, then $\tau$ can be chosen so that $\tau(\alpha) = \alpha'$.

  We discussed $n$-th roots of unity (not a comprehensive study of cyclotomic extensions, but rather a reminder of basic facts). 

  We defined normal extensions and proved that if $K/F$ is normal if and only if any irreducible $P \in F[x]$ that has a root in $K$ splits in $K$. We discussed the significance of normal extensions in the context of the Isomorphism Extension Theorem: an extension of $\mathrm{id} : F \rightarrow F$ to $K$ is not only an embedding of $K$ to $\overline{F}$, it is an automorphism of $K$.

  Reading: pages 12-13 from the notes.

  Reading: section 2.1 form the notes.
• Select section 16 March - 22 March

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  16 March - 22 March

  We solved problem set 3.

  We solved the sample midterm.

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    Problem Set 4 File
• Select section 23 March - 29 March

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  23 March - 29 March

  On Monday we had our midterm exam.

  On Friday, we discussed the notion of separability.

  Reading: pages 15-17 from the notes.
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  30 March - 5 April

  Current week
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  13 April - 19 April
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  20 April - 26 April
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  11 May - 17 May
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  18 May - 24 May